Content check — Calculus Vol. 1, Chapter 1: Functions and Graphs

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 1: Functions and Graphs

199 problems · 45 figures

Checkpoint 1.1 §1.1 source ↗

For

f (x) =

x^{2} -

3 x +

5,

evaluate

f (1)

and

f (a + h) .

Answer

f (1) = 3

and

f (a + h) =

a^{2} +

2 a h +

h^{2} -

3 a -

3 h +

5

Checkpoint 1.2 §1.1 source ↗

Find the domain and range for

f (x) =

\sqrt{4 - 2 x} +

5 .

Answer

Domain =

{x | x \leq 2},

range =

{y | y \geq 5}

Checkpoint 1.3 §1.1 source ↗

Find the zeros of

f (x) =

x^{3} -

5 x^{2} +

6 x .

Answer

x = 0, 2, 3

Checkpoint 1.4 §1.1 source ↗

For

f (x) = x^{2} + 3

and

g (x) = 2 x - 5,

find

(f / g) (x)

and state its domain.

Answer

(\frac{f}{g}) (x) =

\frac{x^{2} + 3}{2 x - 5} .

The domain is

{x | x \neq \frac{5}{2}} .

Checkpoint 1.5 §1.1 source ↗

Let

f (x) = 2 - 5 x .

Let

g (x) = \sqrt{x} .

Find

(f \circ g) (x) .

Answer

(f \circ g) (x) =

2 -

5 \sqrt{x} .

Checkpoint 1.6 §1.1 source ↗

If items are on sale for

10 %

off their original price, and a customer has a coupon for an additional

30 %

off, what will be the final price for an item that is originally

x

dollars, after applying the coupon to the sale price?

Answer

(g \circ f) (x) =

0.63 x

Checkpoint 1.7 §1.1 source ↗

Determine whether

f (x) = 4 x^{3} - 5 x

is even, odd, or neither.

Answer

f (x)

is odd.

Checkpoint 1.8 §1.1 source ↗

For the function

f (x) =

| x +

2 | -

4,

find the domain and range.

Answer

Domain =

(− \infty, \infty),

range =

{y | y \geq −4} .

1-1 §1.1 source ↗

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

$x$	$y$	$x$	$y$
−3	9	1	1
−2	4	2	4
−1	1	3	9
0	0

Answer

a. Domain =

{−3, −2, −1, 0, 1, 2, 3},

range =

{0, 1, 4, 9}

b. Yes, a function

1-3 §1.1 source ↗

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

$x$	$y$	$x$	$y$
1	−3	1	1
2	−2	2	2
3	−1	3	3
0	0

Answer

a. Domain =

{0, 1, 2, 3},

range =

{−3, −2, −1, 0, 1, 2, 3}

b. No, not a function

1-5 §1.1 source ↗

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

$x$	$y$	$x$	$y$
3	3	15	1
5	2	21	2
8	1	33	3
10	0

Answer

a. Domain =

{3, 5, 8, 10, 15, 21, 33},

range =

{0, 1, 2, 3}

b. Yes, a function

1-7 §1.1 source ↗

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f (0)$ b. $f (1)$ c. $f (3)$ d. $f (− x)$ e. $f (a)$ f. $f (a + h)$

$f (x) = 5 x - 2$

Answer

−2

b. 3 c. 13 d.

−5 x - 2

5 a - 2

5 a + 5 h - 2

1-9 §1.1 source ↗

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f (0)$ b. $f (1)$ c. $f (3)$ d. $f (− x)$ e. $f (a)$ f. $f (a + h)$

$f (x) = \frac{2}{x}$

Answer

a. Undefined b. 2 c.

\frac{2}{3}

- \frac{2}{x}

\frac{2}{a}

\frac{2}{a + h}

1-11 §1.1 source ↗

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f (0)$ b. $f (1)$ c. $f (3)$ d. $f (− x)$ e. $f (a)$ f. $f (a + h)$

$f (x) = \sqrt{6 x + 5}$

Answer

\sqrt{5}

\sqrt{11}

\sqrt{23}

\sqrt{−6 x + 5}

\sqrt{6 a + 5}

\sqrt{6 a + 6 h + 5}

1-13 §1.1 source ↗

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f (0)$ b. $f (1)$ c. $f (3)$ d. $f (− x)$ e. $f (a)$ f. $f (a + h)$

$f (x) = 9$

Answer

a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

1-15 §1.1 source ↗

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

$g (x) = \sqrt{8 x - 1}$

Answer

x \geq \frac{1}{8};

y \geq 0;

x =

\frac{1}{8};

no y-intercept

1-17 §1.1 source ↗

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

$f (x) = −1 + \sqrt{x + 2}$

Answer

x \geq −2;

y \geq −1;

x =

−1;

y =

−1 +

\sqrt{2}

1-19 §1.1 source ↗

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

$g (x) = \frac{3}{x - 4}$

Answer

x \neq 4; y \neq 0;

no x-intercept;

y = - \frac{3}{4}

1-21 §1.1 source ↗

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

$g (x) = \sqrt{\frac{7}{x - 5}}$

Answer

x >

5;

y >

0;

no intercepts

1-23 §1.1 source ↗

For the following exercises, sketch the graph with the aid of the tables given:

$f (x) = 3 x - 6$

$x$	$y$	$x$	$y$
−3	−15	1	−3
−2	−12	2	0
−1	−9	3	3
0	−6

Answer

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 3. The graph is of the function “f(x) = 3x - 6”, which is an increasing straight line. The function has an x intercept at (2, 0) and the y intercept is not shown.

1-25 §1.1 source ↗

For the following exercises, sketch the graph with the aid of the tables given:

$f (x) = 2 | x |$

$x$	$y$	$x$	$y$
−3	6	1	2
−2	4	2	4
−1	2	3	6
0	0

Answer

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -2 to 6. The graph is of the function “f(x) = 2 times the absolute value of x”. The function decreases in a straight line until it hits the origin, then begins to increase in a straight line. The function x intercept and y intercept are at the origin.

1-27 §1.1 source ↗

For the following exercises, sketch the graph with the aid of the tables given:

$f (x) = x^{3}$

$x$	$y$	$x$	$y$
−3	−27	1	1
−2	−8	2	8
−1	−1	3	27
0	0

Answer

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -27 to 27. The graph is of the function “f(x) = x cubed”. The curved function increases until it hits the origin, where it levels out and then becomes even. After the origin the graph begins to increase again. The x intercept and y intercept are both at the origin.

1-29 §1.1 source ↗

For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.

Answer

Function; a. Domain: all real numbers, range:

y \geq 0

x = \pm 1

y = 1

−1 <

x <

0

and

1 < x < \infty

− \infty <

x <

-

1

and

0 < x < 1

f. Not constant g. y-axis h. Even

1-31 §1.1 source ↗

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.

Answer

Function; a. Domain: all real numbers, range:

−1.5 \leq y \leq 1.5

x = 0

y = 0

all real numbers

e. None f. Not constant g. Origin h. Odd

1-33 §1.1 source ↗

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.

Answer

Function; a. Domain:

− \infty <

x <

\infty,

range:

−2 \leq y \leq 2

x = 0

y = 0

−2 <

x <

2

e. Not decreasing f.

− \infty <

x <

-

2

and

2 < x < \infty

g. Origin h. Odd

1-35 §1.1 source ↗

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).

Answer

Function; a. Domain:

−4 \leq x \leq 4,

range:

−4 \leq y \leq 4

x = 1.2

y = 4

d. Not increasing e.

0 < x < 4

−4 <

x <

0

g. No Symmetry h. Neither

1-37 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $f + g$ b. $f - g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.

$f (x) =$ $x -$ $8,$ $g (x) =$ $5 x^{2}$

Answer

5 x^{2} + x - 8;

all real numbers b.

−5 x^{2} + x - 8;

all real numbers c.

5 x^{3} - 40 x^{2};

all real numbers d.

\frac{x - 8}{5 x^{2}}; x \neq 0

1-39 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $f + g$ b. $f - g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.

$f (x) =$ $9 -$ $x^{2},$ $g (x) =$ $x^{2} -$ $2 x -$ $3$

Answer

−2 x + 6;

all real numbers b.

−2 x^{2} + 2 x + 12;

all real numbers c.

− x^{4} +

2 x^{3} +

12 x^{2} -

18 x -

27;

all real numbers d.

-

\frac{x + 3}{x + 1};

x \neq − 1,

3

1-41 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $f + g$ b. $f - g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.

$f (x) =$ $6 +$ $\frac{1}{x},$ $g (x) =$ $\frac{1}{x}$

Answer

6 + \frac{2}{x}; x \neq 0

b. 6;

x \neq 0

\frac{6}{x} + \frac{1}{x^{2}}; x \neq 0

6 x + 1; x \neq 0

1-43 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

$f (x) =$ $x +$ $4,$ $g (x) =$ $4 x -$ $1$

Answer

4 x + 3;

all real numbers b.

4 x + 15;

all real numbers

1-45 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

$f (x) =$ $x^{2} +$ $7,$ $g (x) =$ $x^{2} -$ $3$

Answer

x^{4} - 6 x^{2} + 16;

all real numbers b.

x^{4} + 14 x^{2} + 46;

all real numbers

1-47 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

$f (x) =$ $\frac{3}{2 x + 1},$ $g (x) =$ $\frac{2}{x}$

Answer

\frac{3 x}{4 + x};

x \neq 0,

−4

\frac{4 x + 2}{3}; x \neq - \frac{1}{2}

1-49 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

The table below lists the NBA championship winners for the years 2001 to 2012.

Year	Winner
2001	LA Lakers
2002	LA Lakers
2003	San Antonio Spurs
2004	Detroit Pistons
2005	San Antonio Spurs
2006	Miami Heat
2007	San Antonio Spurs
2008	Boston Celtics
2009	LA Lakers
2010	LA Lakers
2011	Dallas Mavericks
2012	Miami Heat

a. Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.

b. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.

Answer

a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.

1-51 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

[T] The volume of a cube depends on the length of the sides $s .$

a. Write a function $V (s)$ for the volume of a cube.

b. Find and interpret $V (11.8) .$

Answer

V (s) = s^{3}

V (11.8) \approx

1643;

a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

1-53 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

[T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank.

a. Write a formula that models this situation.

b. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.

c. Determine the domain and range of the function.

d. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

Answer

N (x) = 15 x

b. i.

N (20) =

15 (20) =

300;

therefore, the vehicle can travel 300 mi on a full tank of gas. Ii.

N (15) = 225;

therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain:

0 \leq x \leq 20;

range:

[0, 300]

d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

1-55 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

[T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r (t) =$ $6 -$ $[5 / (t^{2} + 1)],$ where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

a. Express the area of the bacteria as a function of time.

b. Find the exact and approximate area of the bacterial culture in 3 hours.

c. Express the circumference of the bacteria as a function of time.

d. Find the exact and approximate circumference of the bacteria in 3 hours.

Answer

A (t) =

A (r (t)) =

π \cdot {(6 - \frac{5}{t^{2} + 1})}^{2}

b. Exact:

\frac{121 π}{4};

approximately 95 cm² c.

C (t) =

C (r (t)) =

2 π (6 - \frac{5}{t^{2} + 1})

d. Exact:

11 π;

approximately 35 cm

1-57 §1.1 source ↗

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$ and b. $(g \circ f) (x)$ Simplify the results. Find the domain of each of the results.

[T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell.

a. Write a function $y = S (x)$ that models a worker’s monthly salary based on the number of skateboards x he or she sells.

b. Find the monthly salary when a worker sells 25, 40, or 55 skateboards.

c. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line $y = 1400 .)$

An image of a graph. The y axis runs from 0 to 1800 and the x axis runs from 0 to 100. The graph is of the function “y = s(x)”, which is a increasing straight line. The function has a y intercept at (0, 750) and the x intercept is not shown.

Answer

S (x) =

8.5 x +

750

b. $962.50, $1090, $1217.50 c. 77 skateboards

Checkpoint 1.9 §1.2 source ↗

Consider the line passing through points $(−3, 2)$ and $(1, 4) .$ Find the slope of the line.

Find an equation of that line in point-slope form. Find an equation of that line in slope-intercept form.

Answer

$m = 1 / 2 .$ The point-slope form is

$y -$ $4 =$ $\frac{1}{2} (x - 1) .$

The slope-intercept form is

$y = \frac{1}{2} x + \frac{7}{2} .$

Checkpoint 1.10 §1.2 source ↗

Consider the quadratic function

f (x) =

3 x^{2} -

6 x +

2 .

Find the zeros of

f .

Does the parabola open upward or downward?

Answer

The zeros are

x = 1 \pm \sqrt{3} / 3 .

The parabola opens upward.

Checkpoint 1.11 §1.2 source ↗

Find the domain and range for the function

f (x) =

(5 x + 2) / (2 x - 1) .

Answer

The domain is the set of real numbers

x

such that

x \neq 1 / 2 .

The range is the set

{y | y \neq 5 / 2} .

Checkpoint 1.12 §1.2 source ↗

Find the domain for each of the following functions:

f (x) =

(5 - 2 x) / (x^{2} + 2)

and

g (x) = \sqrt{5 x - 1} .

Answer

The domain of

f

(−∞, ∞).

The domain of

g

{x | x \geq 1 / 5} .

Checkpoint 1.13 §1.2 source ↗

f (x) = x / 2

an algebraic or a transcendental function?

Answer

Algebraic

Checkpoint 1.14 §1.2 source ↗

Sketch a graph of the function

f (x) = {\begin{matrix} 2 - x, x \leq 2 \\ x + 2, x > 2 \end{matrix} .

Answer

An image of a graph. The x axis runs from -6 to 5 and the y axis runs from -2 to 7. The graph is of a function that has two pieces. The first piece is a decreasing line that ends at the closed circle point (2, 0) and has the label “f(x) = 2 - x, for x ≤ 2. The second piece is an increasing line and begins at the open circle point (2, 4) and has the label “f(x) = x + 2, for x greater than 2.The function has an x intercept at (2, 0) and a y intercept at (0, 2).

Checkpoint 1.15 §1.2 source ↗

The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is

49 ¢

for the first ounce and

21 ¢

for each additional ounce. Write a piecewise-defined function describing the cost

C

as a function of the weight

x

for

0 < x \leq 3,

where

C

is measured in cents and

x

is measured in ounces.

Answer

C (x) = {\begin{matrix} 49, 0 < x \leq 1 \\ 70, 1 < x \leq 2 \\ 91, 2 < x \leq 3 \end{matrix}

Checkpoint 1.16 §1.2 source ↗

Describe how the function

f (x) =

− {(x + 1)}^{2} -

4

can be graphed using the graph of

y = x^{2}

and a sequence of transformations.

Answer

Shift the graph

y = x^{2}

to the left 1 unit, reflect about the

x

-axis, then shift down 4 units.

1-59 §1.2 source ↗

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

$(−2, 4)$ and $(1, 1)$

Answer

a. −1 b. Decreasing

1-61 §1.2 source ↗

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

$(3, 5)$ and $(−1, 2)$

Answer

a. 3/4 b. Increasing

1-63 §1.2 source ↗

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

$(2, 3)$ and $(5, 7)$

Answer

a. 4/3 b. Increasing

1-65 §1.2 source ↗

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

$(2, 4)$ and $(1, 4)$

Answer

a. 0 b. Horizontal

1-67 §1.2 source ↗

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.

Slope $= −6,$ passes through $(1, 3)$

Answer

y = −6 x + 9

1-69 §1.2 source ↗

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.

Slope $= \frac{1}{3},$ passes through $(0, 4)$

Answer

y = \frac{1}{3} x + 4

1-71 §1.2 source ↗

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.

Passing through $(2, 1)$ and $(−2, −1)$

Answer

y = \frac{1}{2} x

1-73 §1.2 source ↗

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.

$x$ -intercept $= 5$ and $y$ -intercept $= −3$

Answer

y = \frac{3}{5} x - 3

1-75 §1.2 source ↗

For the following exercises, for each linear equation, a. give the slope $m$ and $y$ -intercept b, if any, and b. graph the line.

$y = 2 x - 3$

Answer

a. $(m = 2, b = −3)$ b.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows an increasing straight line function with a y intercept at (0, -3) and a x intercept at (1.5, 0).

1-77 §1.2 source ↗

For the following exercises, for each linear equation, a. give the slope $m$ and $y$ -intercept b, if any, and b. graph the line.

$f (x) = −6 x$

Answer

a. $(m = −6, b = 0)$ b.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a decreasing straight line function with a y intercept and x intercept both at the origin. There is an unlabeled point on the function at (0.5, -3).

1-79 §1.2 source ↗

For the following exercises, for each linear equation, a. give the slope $m$ and $y$ -intercept b, if any, and b. graph the line.

$4 y + 24 = 0$

Answer

a. $(m = 0, b = −6)$ b.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -7 to 1. The graph shows a horizontal straight line function with a y intercept at (0, -6) and no x intercept.

1-81 §1.2 source ↗

For the following exercises, for each linear equation, a. give the slope $m$ and $y$ -intercept b, if any, and b. graph the line.

$2 x + 3 y = 6$

Answer

a. $(m = - \frac{2}{3}, b = 2)$ b.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph shows a decreasing straight line function with a y intercept at (0, 2) and a x intercept at (3, 0).

1-83 §1.2 source ↗

For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the $y$ -intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.

$f (x) =$ $2 x^{2} -$ $3 x -$ $5$

Answer

a. 2 b.

\frac{5}{2}, −1;

c. −5 d. As

x \to \pm \infty, y \to \infty

e. Neither

1-85 §1.2 source ↗

$f (x) = \frac{1}{2} x^{2} - 1$

Answer

a. 2 b.

\pm \sqrt{2}

c. −1 d. As

x \to \pm \infty, y \to \infty

e. Even

1-87 §1.2 source ↗

$f (x) = 3 x - x^{3}$

Answer

a. 3 b. 0,

\pm \sqrt{3}

c. 0 d. As

x \to - \infty, y \to \infty

and

x \to \infty, y \to - \infty

e. Odd

1-89 §1.2 source ↗

For the following exercises, use the graph of $f (x) = x^{2}$ to graph each transformed function $g .$

$g (x) =$ ${(x + 3)}^{2} +$ $1$

Answer

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a parabolic function that decreases until the point (-3, 1), then begins increasing. The y intercept is not shown and there are no x intercepts. There are two unplotted points at (-4, 2) and (-2, 2).

1-91 §1.2 source ↗

For the following exercises, use the graph of $f (x) = \sqrt{x}$ to graph each transformed function $g .$

$g (x) = − \sqrt{x} - 1$

Answer

An image of a graph. The x axis runs from -5 to 20 and the y axis runs from -8 to 2. The graph shows a curved function that begins at the point (0, -1), then begins decreasing. The y intercept is at (0, -1) and there is no x intercept. There is an unplotted point at (9, -4).

1-93 §1.2 source ↗

For the following exercises, use the graph of $y = f (x)$ to graph each transformed function $g .$

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a function that starts at point (-3, 0), where it begins to increase until the point (-1, 2). After the point (-1, 2), the function becomes a horizontal line and stays that way until the point (1, 2). After the point (1, 2), the function begins to decrease until the point (3, 0), where the function ends.

$g (x) =$ $f (x - 1) +$ $2$

Answer

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a function that starts at point (-2, 2), where it begins to increase until the point (0, 4). After the point (0, 4), the function becomes a horizontal line and stays that way until the point (2, 4). After the point (2, 4), the function begins to decrease until the point (4, 2), where the function ends.

1-95 §1.2 source ↗

For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.

$f (x) = {\begin{matrix} x^{2} - 3, x < 0 \\ 4 x - 3, x \geq 0 \end{matrix}; f (−4); f (0); f (2)$

Answer

a. $13, −3, 5$ b.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a function that has two pieces. The first piece is a decreasing curve that ends at the point (0, -3). The second piece is an increasing line that begins at the point (0, -3). The function has a x intercepts at the approximate point (1.7, 0) and the point (0.75, 0) and a y intercept at (0, -3).

1-97 §1.2 source ↗

For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.

$g (x) = {\begin{cases} \frac{3}{x - 2}, x \neq 2 \\ 4, x = 2 \end{cases}; g (0); g (−4); g (2)$

Answer

a. $\frac{−3}{2}, \frac{−1}{2}, 4$ b.

An image of a graph. The x axis runs from -10 to 10 and the y axis runs from -10 to 10. The graph is of a function that begins slightly below the x axis and begins to decrease. As the function approaches the unplotted vertical line of “x = 2”, it decreases at a faster rate but never reaches the line “x = 2”. On the right side of the unplotted line “x = 2”, the function starts at the top of graph and begins decreasing and approaches the unplotted horizontal line “y = 0”, but never reaches “y = 0”. There function also includes a plotted point at (2, 4). There is a y intercept at (0, -1.5) and no x intercept.

1-99 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

$g (x) = \sqrt[3]{x}$ is an odd root function

Answer

True;

n = 3

1-101 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

A function of the form $f (x) = x^{b},$ where $b$ is a real valued constant, is an exponential function.

Answer

False;

f (x) = x^{b},

where

b

is a real-valued constant, is a power function

1-103 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

[T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.

a. Find a function $y = V (t)$ that determines the value V of the equipment at the end of t years.

b. Find and interpret the meaning of the $x$ - and $y$ -intercepts for this situation.

c. What is the value of the equipment at the end of 5 years?

d. When will the value of the equipment be $3000?

Answer

V (t) =

−2733 t +

20500

(0, 20, 500)

means that the initial purchase price of the equipment is $20,500;

(7.5, 0)

means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years

1-105 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost $C$ as a function of number of cupcakes made.

a. Find a linear function that relates cost C to x, the number of cupcakes made.

b. Find the cost to bake 160 cupcakes.

c. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)

Answer

C = 0.75 x + 125

b. $245 c. 167 cupcakes

1-107 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

[T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year.

a. Find a linear function that models the value V of the car after t years.

b. Find and interpret $V (4) .$

Answer

V (t) =

−1500 t +

26,000

b. In 4 years, the value of the car is $20,000.

1-109 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

[T] The total cost C to produce a certain item is modeled by the function $C (x) =$ $10.50 x +$ $28,500,$ where x is the number of items produced. Determine the cost to produce 175 items.

Answer

$30,337.50

1-111 §1.2 source ↗

For the following exercises, determine whether the statement is true or false. Explain why.

[T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function $P (t) =$ $1.8576 t +$ $68.052,$ where t is time in years and $t = 0$ corresponds to the beginning of 2000. Use the model to predict the approximate percentage output at the beginning of 2015.

Answer

96% of the total capacity

Checkpoint 1.17 §1.3 source ↗

Express

210 °

using radians. Express

11 π / 6

rad using degrees.

Answer

7 π / 6;

330°

Checkpoint 1.18 §1.3 source ↗

Evaluate

cos (3 π / 4)

and

sin (− π / 6) .

Answer

cos (3 π / 4) =

− \sqrt{2} / 2;

sin (− π / 6) =

−1 / 2

Checkpoint 1.19 §1.3 source ↗

A house painter wants to lean a

20

-ft ladder against a house. If the angle between the base of the ladder and the ground is to be

60 °,

how far from the house should she place the base of the ladder?

Answer

10

Checkpoint 1.20 §1.3 source ↗

Find all solutions to the equation

cos (2 θ) =

sin θ .

Answer

θ =

\frac{3 π}{2} +

2 n π,

\frac{π}{6} +

2 n π,

\frac{5 π}{6} +

2 n π

for

n = 0, \pm 1, \pm 2,…

Checkpoint 1.21 §1.3 source ↗

Prove the trigonometric identity

1 +

{cot}^{2} θ =

{csc}^{2} θ .

Answer

\begin{matrix} 1 + θ & = & 1 + \frac{θ}{θ} \\ = & \frac{θ}{θ} + \frac{θ}{θ} \\ = & \frac{θ + θ}{θ} \\ = & \frac{1}{θ} \\ = & θ \end{matrix}

Checkpoint 1.22 §1.3 source ↗

Describe the relationship between the graph of

f (x) =

3 sin (4 x) -

5

and the graph of

y = sin (x) .

Answer

To graph

f (x) =

3 sin (4 x) -

5,

the graph of

y = sin (x)

needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function

f

will have a period of

π / 2

and an amplitude of 3.

1-113 §1.3 source ↗

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$240 °$

Answer

\frac{4 π}{3} rad

1-115 §1.3 source ↗

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$−60 °$

Answer

\frac{− π}{3}

1-117 §1.3 source ↗

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$330 °$

Answer

\frac{11 π}{6} rad

1-119 §1.3 source ↗

For the following exercises, convert each angle in radians to degrees.

$\frac{7 π}{6} rad$

Answer

210 °

1-121 §1.3 source ↗

For the following exercises, convert each angle in radians to degrees.

$−3 π rad$

Answer

−540 °

1-123 §1.3 source ↗

Evaluate the following functional values.

$cos (\frac{4 π}{3})$

Answer

−0.5

1-125 §1.3 source ↗

Evaluate the following functional values.

$sin (- \frac{3 π}{4})$

Answer

- \frac{\sqrt{2}}{2}

1-127 §1.3 source ↗

Evaluate the following functional values.

$sin (\frac{π}{12})$

Answer

\frac{\sqrt{3} - 1}{2 \sqrt{2}}

1-129 §1.3 source ↗

For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.

An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.

$a = 4, c = 7$

Answer

b = 5.7

sin A =

\frac{4}{7},

cos A =

\frac{5.7}{7},

tan A =

\frac{4}{5.7},

csc A =

\frac{7}{4},

sec A =

\frac{7}{5.7},

cot A =

\frac{5.7}{4}

1-131 §1.3 source ↗

$a =$ $85.3,$ $b =$ $125.5$

Answer

c = 151.7

sin A =

0.5623,

cos A =

0.8273,

tan A =

0.6797,

csc A =

1.778,

sec A =

1.209,

cot A =

1.471

1-133 §1.3 source ↗

$a = 84, b = 13$

Answer

c = 85

sin A =

\frac{84}{85},

cos A =

\frac{13}{85},

tan A =

\frac{84}{13},

csc A =

\frac{85}{84},

sec A =

\frac{85}{13},

cot A =

\frac{13}{84}

1-135 §1.3 source ↗

For the following exercises, $P$ is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle $θ$ with a terminal side that passes through point $P .$ Rationalize denominators.

$P (\frac{7}{25}, y),$ $y >$ $0$

Answer

y = \frac{24}{25}

sin θ =

\frac{24}{25},

cos θ =

\frac{7}{25},

tan θ =

\frac{24}{7},

csc θ =

\frac{25}{24},

sec θ =

\frac{25}{7},

cot θ =

\frac{7}{24}

1-137 §1.3 source ↗

$P (x, \frac{\sqrt{7}}{3}),$ $x <$ $0$

Answer

x = \frac{− \sqrt{2}}{3}

sin θ =

\frac{\sqrt{7}}{3},

cos θ =

\frac{− \sqrt{2}}{3},

tan θ =

\frac{− \sqrt{14}}{2},

csc θ =

\frac{3 \sqrt{7}}{7},

sec θ =

\frac{−3 \sqrt{2}}{2},

cot θ =

\frac{− \sqrt{14}}{7}

1-139 §1.3 source ↗

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

${tan}^{2} x +$ $sin x csc x$

Answer

{sec}^{2} x

1-141 §1.3 source ↗

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$\frac{{tan}^{2} x}{{sec}^{2} x}$

Answer

{sin}^{2} x

1-143 §1.3 source ↗

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

${(1 + tan θ)}^{2} -$ $2 tan θ$

Answer

{sec}^{2} θ

1-145 §1.3 source ↗

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$\frac{cos t}{sin t} +$ $\frac{sin t}{1 + cos t}$

Answer

\frac{1}{sin t} (= csc t)

1-155 §1.3 source ↗

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 sin θ - 1 = 0$

Answer

{\frac{π}{6}, \frac{5 π}{6}}

1-157 §1.3 source ↗

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 {tan}^{2} θ = 2$

Answer

{\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}}

1-159 §1.3 source ↗

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$\sqrt{3} cot θ + 1 = 0$

Answer

{\frac{2 π}{3}, \frac{5 π}{3}}

1-161 §1.3 source ↗

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 cos θ sin θ =$ $sin θ$

Answer

{0, π, \frac{π}{3}, \frac{5 π}{3}}

1-163 §1.3 source ↗

For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0 .$ Write the equation of the graph.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.

Answer

y = 4 sin (\frac{π}{4} x)

1-165 §1.3 source ↗

For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0 .$ Write the equation of the graph.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).

Answer

y = cos (2 π x)

1-167 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

$y = sin (x - \frac{π}{4})$

Answer

a. 1 b.

2 π

\frac{π}{4}

units to the right

1-169 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

$y =$ $\frac{−1}{2} sin (\frac{1}{4} x)$

Answer

\frac{1}{2}

8 π

c. No phase shift

1-171 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

$y =$ $−3 sin (π x + 2)$

Answer

a. 3 b.

2

\frac{2}{π}

units to the left

1-173 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

[T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of $120 °,$ how many inches does it move? Approximate to the nearest whole inch.

Answer

Approximately 42 in.

1-175 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

[T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $ω,$ and is given by $ω = θ / t,$ where $θ$ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. $θ =$ $\frac{7 π}{4} rad,$ $t =$ $10$ sec b. $θ =$ $\frac{3 π}{5} rad,$ $t =$ $8$ sec c. $θ =$ $\frac{2 π}{9} rad,$ $t =$ $1$ min d. $θ =$ $23.76 rad, t =$ $14$ min

Answer

a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min

1-177 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

[T] The area of an isosceles triangle with equal sides of length x is

$\frac{1}{2} x^{2} sin θ,$

where $θ$ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle $θ = 5 π / 12$ rad.

Answer

\approx 30.9 {in}^{2}

1-179 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

[T] An alternating current for outlets in a home has voltage given by the function

$V (t) =$ $150 cos 368 t,$

where V is the voltage in volts at time t in seconds.

a. Find the period of the function and interpret its meaning.

b. Determine the number of periods that occur when 1 sec has passed.

Answer

a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

1-181 §1.3 source ↗

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

[T] Suppose that $T =$ $50 +$ $10 sin [\frac{π}{12} (t - 8)]$ is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.

a. Determine the amplitude and period.

b. Find the temperature 7 hours after midnight.

c. At what time does $T = 60 ° ?$

d. Sketch the graph of $T$ over $0 \leq t \leq 24 .$

Answer

a. Amplitude = $10;$ $period =$ $24$ b. $47.4 ° F$ c. 14 hours later, or 2 p.m. d.

An image of a graph. The x axis runs from 0 to 365 and is labeled “t, hours after midnight”. The y axis runs from 0 to 20 and is labeled “T, degrees in Fahrenheit”. The graph is of a curved wave function that starts at the approximate point (0, 41.3) and begins decreasing until the point (2, 40). After this point, the function increases until the point (14, 60). After this point, the function begins decreasing again.

Checkpoint 1.23 §1.4 source ↗

Is the function $f$ graphed in the following image one-to-one?

An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of the function “f(x) = (x cubed) - x” which is a curved function. The function increases, decreases, then increases again. The x intercepts are at the points (-1, 0), (0,0), and (1, 0). The y intercept is at the origin.

Answer

No.

Checkpoint 1.24 §1.4 source ↗

Find the inverse of the function

f (x) =

3 x / (x - 2) .

State the domain and range of the inverse function.

Answer

f^{−1} (x) =

\frac{2 x}{x - 3} .

The domain of

f^{−1}

{x | x \neq 3} .

The range of

f^{−1}

{y | y \neq 2} .

Checkpoint 1.25 §1.4 source ↗

Sketch the graph of

f (x) = 2 x + 3

and the graph of its inverse using the symmetry property of inverse functions.

Answer

An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of two functions. The first function is “f(x) = 2x +3”, an increasing straight line function. The function has an x intercept at (-1.5, 0) and a y intercept at (0, 3). The second function is “f inverse (x) = (x - 3)/2”, an increasing straight line function, which increases at a slower rate than the first function. The function has an x intercept at (3, 0) and a y intercept at (0, -1.5). In addition to the two functions, there is a diagonal dotted line potted with the equation “y =x”, which shows that “f(x)” and “f inverse (x)” are mirror images about the line “y =x”.

Checkpoint 1.26 §1.4 source ↗

Consider

f (x) = 1 / x^{2}

restricted to the domain

(− \infty, 0) .

Verify that

f

is one-to-one on this domain. Determine the domain and range of the inverse of

f

and find a formula for

f^{−1} .

Answer

The domain of

f^{−1}

(0, \infty) .

The range of

f^{−1}

(− \infty, 0) .

The inverse function is given by the formula

f^{−1} (x) =

−1 / \sqrt{x} .

1-183 §1.4 source ↗

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that decreases in a straight in until the origin, where it begins to increase in a straight line. The x intercept and y intercept are both at the origin.

Answer

Not one-to-one

1-185 §1.4 source ↗

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that resembles a semi-circle, the top half of a circle. The function starts at the point (-3, 0) and increases until the point (0, 3), where it begins decreasing until it ends at the point (3, 0). The x intercepts are at (-3, 0) and (3, 0). The y intercept is at (0, 3).

Answer

Not one-to-one

1-187 §1.4 source ↗

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function that is always increasing. The x intercept and y intercept are both at the origin.

Answer

One-to-one

1-189 §1.4 source ↗

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) =$ $x^{2} -$ $4,$ $x \geq 0$

Answer

f^{−1} (x) = \sqrt{x + 4}

b. Domain

: x \geq −4,

range : y \geq 0

1-191 §1.4 source ↗

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) = x^{3} + 1$

Answer

f^{−1} (x) = \sqrt[3]{x - 1}

b. Domain: all real numbers, range: all real numbers

1-193 §1.4 source ↗

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) = \sqrt{x - 1}$

Answer

f^{−1} (x) =

x^{2} +

1,

b. Domain:

x \geq 0,

range:

y \geq 1

1-195 §1.4 source ↗

For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of an increasing straight line function labeled “f” that is always increasing. The x intercept is at (-2, 0) and y intercept are both at (0, 1).

Answer

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of two functions. The first function is an increasing straight line function labeled “f”. The x intercept is at (-2, 0) and y intercept are both at (0, 1). The second function is of an increasing straight line function labeled “f inverse”. The x intercept is at the point (1, 0) and the y intercept is at the point (0, -2).

1-197 §1.4 source ↗

For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.

An image of a graph. The x axis runs from -8 to 8 and the y axis runs from -8 to 8. The graph is of an increasing straight line function labeled “f”. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1).

Answer

An image of a graph. The x axis runs from 0 to 8 and the y axis runs from 0 to 8. The graph is of two function. The first function is an increasing straight line function labeled “f”. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1). The second function is an increasing straight line function labeled “f inverse”. The function starts at the point (1, 0) and increases in straight line until the point (6, 4). After this point, the function continues to increase, but at a faster rate than before, as it approaches the point (8, 8). The function does not have an y intercept and the x intercept is (1, 0).

1-199 §1.4 source ↗

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) =$ $8 x,$ $g (x) =$ $\frac{x}{8}$

Answer

These are inverses.

1-201 §1.4 source ↗

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) =$ $5 x -$ $7,$ $g (x) =$ $\frac{x + 5}{7}$

Answer

These are not inverses.

1-203 §1.4 source ↗

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) =$ $\frac{1}{x - 1},$ $x \neq 1,$ $g (x) =$ $\frac{1}{x} +$ $1,$ $x \neq 0$

Answer

These are inverses.

1-205 §1.4 source ↗

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) =$ $x^{2} +$ $2 x +$ $1,$ $x \geq −1,$ $g (x) =$ $−1 +$ $\sqrt{x},$ $x \geq 0$

Answer

These are inverses.

1-207 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

${tan}^{−1} (\frac{\sqrt{3}}{3})$

Answer

\frac{π}{6}

1-209 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

${cot}^{−1} (1)$

Answer

\frac{π}{4}

1-211 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

${cos}^{−1} (\frac{\sqrt{3}}{2})$

Answer

\frac{π}{6}

1-213 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

$sin ({cos}^{−1} (\frac{\sqrt{2}}{2}))$

Answer

\frac{\sqrt{2}}{2}

1-215 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

${tan}^{−1} (tan (- \frac{π}{6}))$

Answer

- \frac{π}{6}

1-217 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function $V =$ $f (x) =$ $500 (0.04 - x^{2})$ for $0 \leq x \leq 0.2 .$

a. Find $x = f^{−1} (V) .$

b. Interpret what the inverse function is used for.

c. Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.

Answer

x =

f^{−1} (V) =

\sqrt{0.04 - \frac{V}{500}}

b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. c. 0.1 cm; 0.14 cm; 0.17 cm

1-219 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] The cost to remove a toxin from a lake is modeled by the function

$C (p) =$ $75 p / (85 - p),$ where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.

a. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.

b. Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.

Answer

a. $31,250, $66,667, $107,143 b.

(p = \frac{85 C}{C + 75})

c. 34 ppb

1-221 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $μ =$ $2 {sin}^{−1} (\frac{1}{M}) .$

Find the Mach angle (to the nearest degree) for the following Mach numbers.

An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways “V” shape, with the airplane flying directly into the opening of the “V” shape. The “V” shape is labeled “mach wave”. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the “V” shape. This arrow has the label “velocity = v”. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the “V” shape. This arrow has the label “speed of sound = a”. Between these two arrows is an angle labeled “Mach angle”. There is also text in the image that reads “mach = M greater than 1.0”.

a. $M = 1.4$

b. $M = 2.8$

c. $M = 4.3$

Answer

~ 92 °

~ 42 °

~ 27 °

1-223 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] The average temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

$T (x) =$ $5 +$ $18 sin [\frac{π}{6} (x - 4.6)],$ where $x$ is time in months and $x = 1.00$ corresponds to January 1. Determine the day(s) (month and day) when the average temperature is $21 ° C .$ Use the integer portion of your answer(s) as the month and calculate the day of the month from the decimal portion.

Answer

x \approx

6.69,

8.51;

so, the temperature occurs on June 21 and August 15

1-225 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] An object moving in simple harmonic motion is modeled by the function

$s (t) =$ $−6 cos (\frac{π t}{2}),$ where $s$ is measured in inches and $t$ is measured in seconds. Determine the first time when the distance moved is 4.5 in.

Answer

~ 1.5 sec

1-227 §1.4 source ↗

For the following exercises, evaluate the functions. Give the exact value.

[T] Use a calculator to evaluate ${tan}^{−1} (tan (2.1))$ and ${cos}^{−1} (cos (2.1)) .$ Explain the results of each.

Answer

{tan}^{−1} (tan (2.1)) \approx

-

1.0416;

the expression does not equal 2.1 since

2.1 >

1.57 =

\frac{π}{2}

—in other words, it is not in the restricted domain of

tan x . {cos}^{−1} (cos (2.1)) =

2.1,

since 2.1 is in the restricted domain of

cos x .

Checkpoint 1.27 §1.5 source ↗

Given the exponential function

f (x) =

100 \cdot 3^{x / 2},

evaluate

f (4)

and

f (10) .

Answer

f (4) =

900;

f (10) =

24,

300 .

Checkpoint 1.28 §1.5 source ↗

Use the laws of exponents to simplify

(6 x^{−3} y^{2}) / (12 x^{−4} y^{5}) .

Answer

x / (2 y^{3})

Checkpoint 1.29 §1.5 source ↗

$ 750

is invested in an account at an annual interest rate of

4 %,

compounded continuously, find a formula for the amount of money in the account after

t

years. Find the amount of money after

30

years.

Answer

A (t) =

750 e^{0.04 t} .

After

30

years, there will be approximately

$ 2, 490.09 .

Checkpoint 1.30 §1.5 source ↗

Solve

e^{2 x} / (3 + e^{2 x}) =

1 / 2 .

Answer

x = \frac{ln 3}{2}

Checkpoint 1.31 §1.5 source ↗

Solve

ln (x^{3}) -

4 ln (x) =

1 .

Answer

x = \frac{1}{e}

Checkpoint 1.32 §1.5 source ↗

Use the change-of-base formula and a calculating utility to evaluate

{log}_{4} 6 .

Answer

1.29248

Checkpoint 1.33 §1.5 source ↗

Compare the relative severity of a magnitude

8.4

earthquake with a magnitude

7.4

earthquake.

Answer

The magnitude

8.4

earthquake is roughly

10

times as severe as the magnitude

7.4

earthquake.

Checkpoint 1.34 §1.5 source ↗

Simplify

cosh (2 ln x) .

Answer

(x^{2} + x^{−2}) / 2

Checkpoint 1.35 §1.5 source ↗

Evaluate

{tanh}^{−1} (1 / 2) .

Answer

\frac{1}{2} ln (3) \approx

0.5493 .

1-229 §1.5 source ↗

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

$f (x) = 5^{x}$ a. $x = 3$ b. $x = \frac{1}{2}$ c. $x = \sqrt{2}$

Answer

a. 125 b. 2.24 c. 9.74

1-231 §1.5 source ↗

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

$f (x) = 10^{x}$ a. $x = −2$ b. $x = 4$ c. $x = \frac{5}{3}$

Answer

a. 0.01 b. 10,000 c. 46.42

1-233 §1.5 source ↗

For the following exercises, match the exponential equation to the correct graph.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -2 to 8. The graph is of a decreasing curved function. The function decreases until it approaches the line “y = 2”, but never touches this line. The y intercept is at the point (0, 3) and there is no x intercept.

Answer

1-235 §1.5 source ↗

For the following exercises, match the exponential equation to the correct graph.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, (1/3)). Another point of the graph is at (1, 1).

Answer

1-237 §1.5 source ↗

For the following exercises, match the exponential equation to the correct graph.

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, -1). Another point of the graph is at (-1, -3).

Answer

1-239 §1.5 source ↗

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

$f (x) = e^{x} + 2$

Answer

Domain: all real numbers, range: $(2, \infty), y = 2$

1-241 §1.5 source ↗

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

$f (x) = 3^{x + 1}$

Answer

Domain: all real numbers, range: $(0, \infty), y = 0$

1-243 §1.5 source ↗

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

$f (x) = 1 - 2^{− x}$

Answer

Domain: all real numbers, range: $(− \infty, 1), y = 1$

1-245 §1.5 source ↗

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

$f (x) = e^{− x} - 1$

Answer

Domain: all real numbers, range: $(−1, \infty), y = −1$

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the line “y = -1” without touching it. There x intercept and the y intercept are both at the origin. There is an approximate point on the graph at (-1, 1.7).

1-247 §1.5 source ↗

For the following exercises, write the equation in equivalent exponential form.

${log}_{8} 2 = \frac{1}{3}$

Answer

8^{1 / 3} = 2

1-249 §1.5 source ↗

For the following exercises, write the equation in equivalent exponential form.

${log}_{5} 25 = 2$

Answer

5^{2} = 25

1-251 §1.5 source ↗

For the following exercises, write the equation in equivalent exponential form.

$ln (\frac{1}{e^{3}}) = −3$

Answer

e^{−3} = \frac{1}{e^{3}}

1-253 §1.5 source ↗

For the following exercises, write the equation in equivalent exponential form.

$ln 1 = 0$

Answer

e^{0} = 1

1-255 §1.5 source ↗

For the following exercises, write the equation in equivalent logarithmic form.

$4^{−2} = \frac{1}{16}$

Answer

{log}_{4} (\frac{1}{16}) =

−2

1-257 §1.5 source ↗

For the following exercises, write the equation in equivalent logarithmic form.

$9^{0} = 1$

Answer

{log}_{9} 1 = 0

1-259 §1.5 source ↗

For the following exercises, write the equation in equivalent logarithmic form.

$\sqrt[3]{64} = 4$

Answer

{log}_{64} 4 = \frac{1}{3}

1-261 §1.5 source ↗

For the following exercises, write the equation in equivalent logarithmic form.

$9^{y} = 150$

Answer

{log}_{9} 150 = y

1-263 §1.5 source ↗

For the following exercises, write the equation in equivalent logarithmic form.

$4^{−3 / 2} = 0.125$

Answer

{log}_{4} 0.125 =

-

\frac{3}{2}

1-265 §1.5 source ↗

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

$f (x) =$ $ln (x - 1)$

Answer

Domain: $(1, \infty),$ range: $(− \infty, \infty), x = 1$

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = 1”. There is no y intercept and the x intercept is at the approximate point (2, 0).

1-267 §1.5 source ↗

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

$f (x) = 1 - ln x$

Answer

Domain: $(0, \infty),$ range: $(− \infty, \infty), x = 0$

An image of a graph. The x axis runs from -1 to 9 and the y axis runs from -5 to 5. The graph is of a decreasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (e, 0).

1-269 §1.5 source ↗

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

$f (x) =$ $ln (x + 1)$

Answer

Domain: $(−1, \infty),$ range: $(− \infty, \infty), x = −1$

1-271 §1.5 source ↗

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

${log}_{3} \frac{9 a^{3}}{b}$

Answer

2 +

3 {log}_{3} a -

{log}_{3} b

1-273 §1.5 source ↗

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

${log}_{5} \sqrt{125 x y^{3}}$

Answer

\frac{3}{2} +

\frac{1}{2} {log}_{5} x +

\frac{3}{2} {log}_{5} y

1-275 §1.5 source ↗

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

$ln (\frac{6}{\sqrt{e^{3}}})$

Answer

- \frac{3}{2} + ln 6

1-277 §1.5 source ↗

For the following exercises, solve the exponential equation exactly.

$e^{3 x} - 15 = 0$

Answer

\frac{ln 15}{3}

1-279 §1.5 source ↗

For the following exercises, solve the exponential equation exactly.

$4^{x + 1} - 32 = 0$

Answer

\frac{3}{2}

1-281 §1.5 source ↗

For the following exercises, solve the exponential equation exactly.

$10^{x} = 7.21$

Answer

log 7.21

1-283 §1.5 source ↗

For the following exercises, solve the exponential equation exactly.

$7^{3 x - 2} = 11$

Answer

\frac{2}{3} +

\frac{log 11}{3 log 7}

1-285 §1.5 source ↗

For the following exercises, solve the logarithmic equation exactly, if possible.

${log}_{5} x = −2$

Answer

x = \frac{1}{25}

1-287 §1.5 source ↗

For the following exercises, solve the logarithmic equation exactly, if possible.

$log (2 x - 7) = 0$

Answer

x = 4

1-289 §1.5 source ↗

For the following exercises, solve the logarithmic equation exactly, if possible.

${log}_{6} (x + 9) +$ ${log}_{6} x =$ $2$

Answer

x = 3

1-291 §1.5 source ↗

For the following exercises, solve the logarithmic equation exactly, if possible.

$ln x +$ $ln (x - 2) =$ $ln 4$

Answer

1 + \sqrt{5}

1-293 §1.5 source ↗

For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

${log}_{7} 82$

Answer

(\frac{log 82}{log 7} \approx 2.2646)

1-295 §1.5 source ↗

${log}_{0.5} 211$

Answer

(\frac{log 211}{log 0.5} \approx - 7.7211)

1-297 §1.5 source ↗

${log}_{0.2} 0.452$

Answer

(\frac{log 0.452}{log 0.2} \approx 0.4934)

1-299 §1.5 source ↗

[T] The number of bacteria N in a culture after t days can be modeled by the function $N (t) =$ $1300 \cdot {(2)}^{t / 4} .$ Find the number of bacteria present after 15 days.

Answer

~ 17, 491

1-301 §1.5 source ↗

[T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by $A (t) =$ $100,000 \cdot e^{0.055 t} .$ Find the amount A accumulated in 5 years.

Answer

Approximately $131,653 is accumulated in 5 years.

1-303 §1.5 source ↗

[T] The concentration of hydrogen ions in a substance is denoted by $[H^{+}],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $pH =$ $− log [H^{+}] .$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

a. Find the pH of the following substances. Round answers to one digit.

b. Determine whether the substance is an acid or a base.

i. Eggs: $[H^{+}] =$ $1.6 \times 10^{−8}$ mol/L

ii. Beer: $[H^{+}] =$ $3.16 \times 10^{−3}$ mol/L

iii. Tomato Juice: $[H^{+}] =$ $7.94 \times 10^{−5}$ mol/L

Answer

i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

1-305 §1.5 source ↗

[T] According to the World Bank, at the end of 2013 ( $t = 0$ ) the U.S. population was 316 million and was increasing according to the following model:

$P (t) =$ $316 e^{0.0074 t},$ where P is measured in millions of people and t is measured in years after 2013.

a. Based on this model, what will be the population of the United States in 2020?

b. Determine when the U.S. population will be twice what it is in 2013.

Answer

~ 333

million b. 94 years from 2013, or in 2107

1-307 §1.5 source ↗

[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

a. Use the exponential function $Q = Q_{0} e^{k t}$ to determine the value $k,$ which is the growth rate of the bacteria. Round to four decimal places.

b. Determine approximately how long it takes for 200,000 bacteria to grow.

Answer

k \approx 0.0578

\approx 92

hours

1-309 §1.5 source ↗

[T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more intense was the San Francisco earthquake than the Japanese earthquake? See the definition of Richter Scale in [link] in this section.

Answer

The San Francisco earthquake was

10^{3.4} or \approx

2512

times more intense than the Japanese earthquake.

1-311 §1.R source ↗

True or False? Justify your answer with a proof or a counterexample.

$f \circ g = g \circ f,$ assuming f and g are functions.

Answer

False

1-313 §1.R source ↗

True or False? Justify your answer with a proof or a counterexample.

A relation passing the horizontal line test is a function.

Answer

False

1-315 §1.R source ↗

For the following problems, state the domain and range of the given functions:

$f =$ $x^{2} +$ $2 x -$ $3,$ $g =$ $ln (x - 5),$ $h =$ $\frac{1}{x + 4}$

Answer

Domain:

x > 5,

range: all real numbers

1-317 §1.R source ↗

For the following problems, state the domain and range of the given functions:

$f =$ $x^{2} +$ $2 x -$ $3,$ $g =$ $ln (x - 5),$ $h =$ $\frac{1}{x + 4}$

$g \circ f$

Answer

Domain:

x > 2

and

x < - 4,

range: all real numbers

1-319 §1.R source ↗

Find the degree, y-intercept, and zeros for the following polynomial functions.

$f (x) =$ $x^{3} +$ $2 x^{2} -$ $2 x$

Answer

Degree of 3,

y

-intercept: 0, zeros: 0,

\sqrt{3} - 1, −1 - \sqrt{3}

1-321 §1.R source ↗

Simplify the following trigonometric expressions.

${cos}^{2} x - {sin}^{2} x$

Answer

\begin{array}{l} \cos^{2} x - \sin^{2} x = \cos 2 x \\ = 1 - 2 \sin^{2} x \\ = 2 \cos^{2} x - 1 \end{array}

1-323 §1.R source ↗

Solve the following trigonometric equations on the interval $x = [−2 π, 2 π]$ exactly.

${sec}^{2} x -$ $2 sec x +$ $1 =$ $0$

Answer

0, \pm 2 π

1-325 §1.R source ↗

Solve the following logarithmic equations.

${log}_{2} (x + 4) = 3$

Answer

1-327 §1.R source ↗

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse $f^{−1} (x)$ of the function. Justify your answer.

$f (x) = \frac{1}{x}$

Answer

One-to-one; yes, the function has an inverse; inverse:

f^{−1} (x) = \frac{1}{y}

1-329 §1.R source ↗

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

$f (x) =$ $x^{2} +$ $3 x +$ $4$

Answer

x \geq -

\frac{3}{2},

f^{−1} (x) =

-

\frac{3}{2} +

\frac{1}{2} \sqrt{4 y - 7}

1-331 §1.R source ↗

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.

a. Find the equation $C = f (x)$ that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

Answer

C (x) = 300 + 7 x

b. 100 shirts

1-333 §1.R source ↗

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

The population can be modeled by $P (t) =$ $82.5 -$ $67.5 cos [(π / 6) t],$ where $t$ is time in months $(t = 0$ represents January 1) and $P$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

Answer

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

1-335 §1.R source ↗

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation $y = e^{r t},$ where $y$ is the ratio of radiocarbon still present in the material, $t$ is the number of years passed, and $r =$ $−0.0001210$ is the decay rate of radiocarbon.

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

Answer

78.51%

Loading chapter 2…

Problems and answers are adapted from Calculus Volume 1 by Gilbert Strang and Edwin “Jed” Herman, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.