Chapter 3

Try It

3.1 Functions and Function Notation

ⓐyes
ⓑ yes (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

$w = f (d)$

yes

$g (5) = 1$

$m = 8$

$y = f (x) = \frac{\sqrt[3]{x}}{2}$

$g (1) = 8$

$x = 0$ or $x = 2$

ⓐ yes, because each bank account has a single balance at any given time;
ⓑ no, because several bank account numbers may have the same balance;
ⓒ no, because the same output may correspond to more than one input.

10.

ⓐ Yes, letter grade is a function of percent grade;
ⓑ No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

11.

yes

12.

No, because it does not pass the horizontal line test.

3.2 Domain and Range

${- 5, 0, 5, 10, 15}$

$(- \infty, \infty)$

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

$[- \frac{5}{2}, \infty)$

ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3
ⓑ ${x | x \leq - 2 or - 1 \leq x < 3}$
ⓒ $(- \infty, - 2] \cup [- 1, 3)$

domain =[1950,2002] range = [47,000,000,89,000,000]

domain: $(- \infty, 2];$ range: $(- \infty, 0]$

This graph shows a piecewise function with a curve extending left, a horizontal line segment, and a line segment extending right. Open and closed circles mark function behavior at specific points, indicating discontinuities.

3.3 Rates of Change and Behavior of Graphs

$\frac{$ 2.84 - $ 2.31}{5 years} = \frac{$ 0.53}{5 years} = $ 0.106$ per year.

$\frac{1}{2}$

$a + 7$

The local maximum appears to occur at $(- 1, 28),$ and the local minimum occurs at $(5, - 80) .$ The function is increasing on $(- \infty, - 1) \cup (5, \infty)$ and decreasing on $(- 1, 5) .$

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

3.4 Composition of Functions

$\begin{array}{l} (f g) (x) = f (x) g (x) = (x - 1) (x^{2} - 1) = x^{3} - x^{2} - x + 1 \\ (f - g) (x) = f (x) - g (x) = (x - 1) - (x^{2} - 1) = x - x^{2} \end{array}$

No, the functions are not the same.

A gravitational force is still a force, so $a (G (r))$ makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but $G (a (F))$ does not make sense.

$f (g (1)) = f (3) = 3$ and $g (f (4)) = g (1) = 3$

$g (f (2)) = g (5) = 3$

ⓐ 8
ⓑ 20

$[- 4, 0) \cup (0, \infty)$

Possible answer:

$\begin{array}{l} g (x) = \sqrt{4 + x^{2}} \\ h (x) = \frac{4}{3 - x} \\ f = h \circ g \end{array}$

3.5 Transformation of Functions

b (t) = h (t) + 10 = - 4.9 t^{2} + 30 t + 10

The graphs of $f (x)$ and $g (x)$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

Graph of a square root function and a horizontally shift square foot function.

$g (x) = \frac{1}{x - 1} + 1$

ⓐ

$g (x) = - f (x)$

$x$	-2	0	2	4
$g (x)$	$- 5$	$- 10$	$- 15$	$- 20$

ⓑ

$h (x) = f (− x)$

$x$	-2	0	2	4
$h (x)$	15	10	5	unknown

Notice: $g (x) = f (- x)$ looks the same as $f (x)$ .

even

$x$	2	4	6	8
$g (x)$	9	12	15	0

10.

$g (x) = 3 x - 2$

11.

$g (x) = f (\frac{1}{3} x)$ so using the square root function we get $g (x) = \sqrt{\frac{1}{3} x}$

3.6 Absolute Value Functions

using the variable $p$ for passing, $| p - 80 | \leq 20$

$f (x) = - | x + 2 | + 3$

$x = - 1$ or $x = 2$

3.7 Inverse Functions

$h (2) = 6$

Yes

The domain of function $f^{- 1}$ is $(- \infty, - 2)$ and the range of function $f^{- 1}$ is $(1, \infty) .$

ⓐ $f (60) = 50.$ In 60 minutes, 50 miles are traveled.
ⓑ $f^{- 1} (60) = 70.$ To travel 60 miles, it will take 70 minutes.

ⓐ 3
ⓑ 5.6

$x = 3 y + 5$

$f^{- 1} (x) = {(2 - x)}^{2};$ $domain of f : [0, \infty);$ $domain of f^{- 1} : (- \infty, 2]$

3.1 Section Exercises

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

function

11.

function

13.

function

15.

function

17.

function

19.

function

21.

function

23.

function

25.

not a function

27.

$f (- 3) = - 11;$
$f (2) = - 1;$
$f (- a) = - 2 a - 5;$
$- f (a) = - 2 a + 5;$
$f (a + h) = 2 a + 2 h - 5$

29.

$f (- 3) = \sqrt{5} + 5;$
$f (2) = 5;$
$f (- a) = \sqrt{2 + a} + 5;$
$- f (a) = - \sqrt{2 - a} - 5;$
$f (a + h) = \sqrt{2 - a - h} + 5$

31.

$f (- 3) = 2;$ $f (2) = 1 - 3 = - 2;$
$f (- a) = | - a - 1 | - | - a + 1 |;$
$- f (a) = - | a - 1 | + | a + 1 |;$
$f (a + h) = | a + h - 1 | - | a + h + 1 |$

33.

$\frac{g (x) - g (a)}{x - a} = x + a + 2, x \neq a$

35.

a. $f (- 2) = 14;$ b. $x = 3$

37.

a. $f (5) = 10;$ b. $x = - 1$ or $x = 4$

39.

ⓐ $f (t) = 6 - \frac{2}{3} t;$
ⓑ $f (- 3) = 8;$
ⓒ $t = 6$

41.

not a function

43.

function

45.

function

47.

function

49.

function

51.

function

53.

ⓐ $f (0) = 1;$
ⓑ $f (x) = - 3, x = - 2$ or $x = 2$

55.

not a function so it is also not a one-to-one function

57.

one-to- one function

59.

function, but not one-to-one

61.

function

63.

function

65.

not a function

67.

$f (x) = 1, x = 2$

69.

$\begin{matrix} f (- 2) = 14; & f (- 1) = 11; & f (0) = 8; & f (1) = 5; & f (2) = 2 \end{matrix}$

71.

$\begin{matrix} f (- 2) = 4; & f (- 1) = 4.414; & f (0) = 4.732; & f (1) = 5; & f (2) = 5.236 \end{matrix}$

73.

$\begin{matrix} f (- 2) = \frac{1}{9}; & f (- 1) = \frac{1}{3}; & f (0) = 1; & f (1) = 3; & f (2) = 9 \end{matrix}$

75.

77.

$[0, 100]$

79.

$[- 0.001, 0 .001]$

81.

$[- 1, 000, 000, 1,000,000]$

83.

$[0, 10]$

85.

$[−0.1, 0.1]$

87.

$[- 100, 100]$

89.

ⓐ $g (5000) = 50;$
ⓑ The number of cubic yards of dirt required for a garden of 100 square feet is 1.

91.

ⓐ The height of a rocket above ground after 1 second is 200 ft.
ⓑ The height of a rocket above ground after 2 seconds is 350 ft.

3.2 Section Exercises

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

There is no restriction on $x$ for $f (x) = \sqrt[3]{x}$ because you can take the cube root of any real number. So the domain is all real numbers, $(- \infty, \infty) .$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $x$ -values are restricted for $f (x) = \sqrt[]{x}$ to nonnegative numbers and the domain is $[0, \infty) .$

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the $x$ -axis and $y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate $- \infty$ or $\infty .$ Combine the graphs to find the graph of the piecewise function.

$(- \infty, \infty)$

$(- \infty, 3]$

11.

$(- \infty, \infty)$

13.

$(- \infty, \infty)$

15.

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

17.

$(- \infty, - 11) \cup (- 11, 2) \cup (2, \infty)$

19.

$(- \infty, - 3) \cup (- 3, 5) \cup (5, \infty)$

21.

$(- \infty, 5)$

23.

$[6, \infty)$

25.

$(- \infty, - 9) \cup (- 9, 9) \cup (9, \infty)$

27.

domain: $(2, 8],$ range $[6, 8)$

29.

domain: $[- 4, 4],$ range: $[0, 2]$

31.

domain: $[- 5, 3),$ range: $[0, 2]$

33.

domain: $(- \infty, 1],$ range: $[0, \infty)$

35.

domain: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6];$ range: $[- 6, - \frac{1}{6}] \cup [\frac{1}{6}, 6]$

37.

domain: $[- 3, \infty);$ range: $[0, \infty)$

39.

domain: $(- \infty, \infty)$

A graph displays a piecewise function with a jump discontinuity at x=1. The function is defined by a line segment ending with an open circle at (1,1) and another line segment starting with a closed circle at (1,2).

41.

domain: $(- \infty, \infty)$

43.

domain: $(- \infty, \infty)$

A piecewise graph: a parabola curving left from an open circle at (0,0), and a line segment starting from a closed circle at (0,2) extending right. It illustrates a jump discontinuity.

45.

domain: $(- \infty, \infty)$

47.

$\begin{matrix} f (- 3) = 1; & f (- 2) = 0; & f (- 1) = 0; & f (0) = 0 \end{matrix}$

49.

$\begin{matrix} f (- 1) = - 4; & f (0) = 6; & f (2) = 20; & f (4) = 34 \end{matrix}$

51.

$\begin{matrix} f (- 1) = - 5; & f (0) = 3; & f (2) = 3; & f (4) = 16 \end{matrix}$

53.

domain: $(- \infty, 1) \cup (1, \infty)$

55.

Graph of the equation from [-0.5, -0.1].

window: $[- 0.5, - 0.1];$ range: $[4, 100]$

window: $[0.1, 0.5];$ range: $[4, 100]$

57.

$[0, 8]$

59.

Many answers. One function is $f (x) = \frac{1}{\sqrt{x - 2}} .$

61.

ⓐ The fixed cost is $500.
ⓑ The cost of making 25 items is $750.
ⓒ The domain is [0, 100] and the range is [500, 1500].

3.3 Section Exercises

Yes, the average rate of change of all linear functions is constant.

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

$4 (b + 1)$

$4 x + 2 h$

11.

$\frac{- 1}{13 (13 + h)}$

13.

$3 h^{2} + 9 h + 9$

15.

$4 x + 2 h - 3$

17.

$\frac{4}{3}$

19.

increasing on $(- \infty, - 2.5) \cup (1, \infty),$ decreasing on $(- 2.5, 1)$

21.

increasing on $(- \infty, 1) \cup (3, 4),$ decreasing on $(1, 3) \cup (4, \infty)$

23.

local maximum: $(- 3, 60),$ local minimum: $(3, - 60)$

25.

absolute maximum at approximately $(7, 150),$ absolute minimum at approximately $(−7.5, −220)$

27.

ⓐ –3000
ⓑ –1250

29.

-4

31.

33.

–0.167

35.

Local minimum at $(3, - 22),$ decreasing on $(- \infty, 3),$ increasing on $(3, \infty)$

37.

Local minimum at $(- 2, - 2),$ decreasing on $(- 3, - 2),$ increasing on $(- 2, \infty)$

39.

Local maximum at $(- 0.39, 5.98),$ local minima at $(- 3.15, - 47.62)$ and $(2.04, - 32.04),$ decreasing on $(- \infty, - 3.15) \cup (- 0.39, 2.04),$ increasing on $(- 3.15, - 0.39) \cup (2.04, \infty)$

41.

43.

$b = 5$

45.

2.7 gallons per minute

47.

approximately –0.6 milligrams per day

3.4 Section Exercises

Find the numbers that make the function in the denominator $g$ equal to zero, and check for any other domain restrictions on $f$ and $g,$ such as an even-indexed root or zeros in the denominator.

Yes. Sample answer: Let $f (x) = x + 1 and g (x) = x - 1.$ Then $f (g (x)) = f (x - 1) = (x - 1) + 1 = x$ and $g (f (x)) = g (x + 1) = (x + 1) - 1 = x .$ So $f \circ g = g \circ f .$

$(f + g) (x) = 2 x + 6,$ domain: $(- \infty, \infty)$

$(f - g) (x) = 2 x^{2} + 2 x - 6,$ domain: $(- \infty, \infty)$

$(f g) (x) = - x^{4} - 2 x^{3} + 6 x^{2} + 12 x,$ domain: $(- \infty, \infty)$

$(\frac{f}{g}) (x) = \frac{x^{2} + 2 x}{6 - x^{2}},$ domain: $(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, \sqrt{6}) \cup (\sqrt{6}, \infty)$

$(f + g) (x) = \frac{4 x^{3} + 8 x^{2} + 1}{2 x},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f - g) (x) = \frac{4 x^{3} + 8 x^{2} - 1}{2 x},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f g) (x) = x + 2,$ domain: $(- \infty, 0) \cup (0, \infty)$

$(\frac{f}{g}) (x) = 4 x^{3} + 8 x^{2},$ domain: $(- \infty, 0) \cup (0, \infty)$

$(f + g) (x) = 3 x^{2} + \sqrt{x - 5},$ domain: $[5, \infty)$

$(f - g) (x) = 3 x^{2} - \sqrt{x - 5},$ domain: $[5, \infty)$

$(f g) (x) = 3 x^{2} \sqrt{x - 5},$ domain: $[5, \infty)$

$(\frac{f}{g}) (x) = \frac{3 x^{2}}{\sqrt{x - 5}},$ domain: $(5, \infty)$

11.

ⓐ $3 = 2 (9 x^{2} - 30 x + 25) + 1 = 18 x^{2} - 60 x + 51$
ⓑ $f (g (x)) = 2 {(3 x - 5)}^{2} + 1$
ⓒ $g (f (x)) = 3 (2 x^{2} + 1) - 5 = 6 x^{2} - 2$
ⓓ $(g \circ g) (x) = 3 (3 x - 5) - 5 = 9 x - 20$
ⓔ $(f \circ f) (- 2) = 163$

13.

$f (g (x)) = \sqrt{x^{2} + 3} + 2, g (f (x)) = x + 4 \sqrt{x} + 7$

15.

$f (g (x)) = \sqrt[3]{\frac{x + 1}{x^{3}}} = \frac{\sqrt[3]{x + 1}}{x}, g (f (x)) = \frac{\sqrt[3]{x} + 1}{x}$

17.

$(f \circ g) (x) = \frac{1}{\frac{2}{x} + 4 - 4} = \frac{x}{2}, (g \circ f) (x) = 2 x - 4$

19.

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

21.

ⓐ $(g \circ f) (x) = - \frac{3}{\sqrt{2 - 4 x}}$
ⓑ $(- \infty, \frac{1}{2})$

23.

ⓐ $(0, 2) \cup (2, \infty);$
ⓑ $(- \infty, - 2) \cup (2, \infty);$
ⓒ $(0, \infty)$

25.

$(1, \infty)$

27.

sample: $\begin{array}{l} f (x) = x^{3} \\ g (x) = x - 5 \end{array}$

29.

sample: $\begin{array}{l} f (x) = \frac{4}{x} \\ g (x) = {(x + 2)}^{2} \end{array}$

31.

sample: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = \frac{1}{2 x - 3} \end{array}$

33.

sample: $\begin{array}{l} f (x) = \sqrt[4]{x} \\ g (x) = \frac{3 x - 2}{x + 5} \end{array}$

35.

sample: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = 2 x + 6 \end{array}$

37.

sample: $\begin{array}{l} f (x) = \sqrt[3]{x} \\ g (x) = (x - 1) \end{array}$

39.

sample: $\begin{array}{l} f (x) = x^{3} \\ g (x) = \frac{1}{x - 2} \end{array}$

41.

sample: $\begin{array}{l} f (x) = \sqrt{x} \\ g (x) = \frac{2 x - 1}{3 x + 4} \end{array}$

43.

45.

47.

49.

51.

53.

55.

57.

59.

61.

63.

65.

67.

69.

71.

73.

$f (g (0)) = 27, g (f (0)) = - 94$

75.

$f (g (0)) = \frac{1}{5}, g (f (0)) = 5$

77.

$18 x^{2} + 60 x + 51$

79.

$g \circ g (x) = 9 x + 20$

81.

83.

$(- \infty, \infty)$

85.

False

87.

$(f \circ g) (6) = 6$ ; $(g \circ f) (6) = 6$

89.

$(f \circ g) (11) = 11, (g \circ f) (11) = 11$

91.

93.

$A (t) = π {(25 \sqrt{t + 2})}^{2}$ and $A (2) = π {(25 \sqrt{4})}^{2} = 2500 π$ square inches

95.

$A (5) = π {(2 (5) + 1)}^{2} = 121 π$ square units

97.

ⓐ $N (T (t)) = 23 {(5 t + 1.5)}^{2} - 56 (5 t + 1.5) + 1$
ⓑ 3.38 hours

3.5 Section Exercises

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

For a function $f,$ substitute $(- x)$ for $(x)$ in $f (x) .$ Simplify. If the resulting function is the same as the original function, $f (- x) = f (x),$ then the function is even. If the resulting function is the opposite of the original function, $f (- x) = - f (x),$ then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

$g (x) = | x - 1 | - 3$

$g (x) = \frac{1}{{(x + 4)}^{2}} + 2$

11.

The graph of $f (x + 43)$ is a horizontal shift to the left 43 units of the graph of $f .$

13.

The graph of $f (x - 4)$ is a horizontal shift to the right 4 units of the graph of $f .$

15.

The graph of $f (x) + 8$ is a vertical shift up 8 units of the graph of $f .$

17.

The graph of $f (x) - 7$ is a vertical shift down 7 units of the graph of $f .$

19.

The graph of $f (x + 4) - 1$ is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of $f .$

21.

decreasing on $(- \infty, - 3)$ and increasing on $(- 3, \infty)$

23.

decreasing on $(0, \infty)$

25.

A graph displays an exponential function h, characterized by a left-to-right increasing curve. It approaches a horizontal asymptote at y=-3 for negative x, and rises steeply for positive x.

27.

29.

31.

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

33.

$f (x) = | x - 3 | - 2$

35.

$f (x) = \sqrt{x + 3} - 1$

37.

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

39.

$f (x) = | x + 3 | - 2$

41.

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

43.

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

45.

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

47.

even

49.

odd

51.

even

53.

The graph of $g$ is a vertical reflection (across the $x$ -axis) of the graph of $f .$

55.

The graph of $g$ is a vertical stretch by a factor of 4 of the graph of $f .$

57.

The graph of $g$ is a horizontal compression by a factor of $\frac{1}{5}$ of the graph of $f .$

59.

The graph of $g$ is a horizontal stretch by a factor of 3 of the graph of $f .$

61.

The graph of $g$ is a horizontal reflection across the $y$ -axis and a vertical stretch by a factor of 3 of the graph of $f .$

63.

$g (x) = | - 4 x |$

65.

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

67.

$g (x) = \frac{1}{2} {(x - 5)}^{2} + 1$

69.

The graph of the function $f (x) = x^{2}$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

71.

The graph of $f (x) = | x |$ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

73.

The graph of the function $f (x) = x^{3}$ is compressed vertically by a factor of $\frac{1}{2} .$

75.

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

77.

The graph of $f (x) = \sqrt{x}$ is shifted right 4 units and then reflected across the vertical line $x = 4.$

79.

81.

3.6 Section Exercises

Isolate the absolute value term so that the equation is of the form $| A | = B .$ Form one equation by setting the expression inside the absolute value symbol, $A,$ equal to the expression on the other side of the equation, $B .$ Form a second equation by setting $A$ equal to the opposite of the expression on the other side of the equation, $- B .$ Solve each equation for the variable.

The graph of the absolute value function does not cross the $x$ -axis, so the graph is either completely above or completely below the $x$ -axis.

The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4.

∣ x − 10 ∣ ≥ 15

There are no x-intercepts.

11.

(−4, 0) and (2, 0)

13.

$(0, - 4), (4, 0), (- 2, 0)$

15.

$(0, 7), (25, 0), (- 7, 0)$

17.

A V-shaped graph with a vertex at (-1, 0), showing points (-3, 2), (-2, 1), (0, 1), and (1, 2).

19.

A V-shaped graph is displayed on a Cartesian coordinate system. The x-axis ranges from -5 to 4, and the y-axis ranges from -3 to 5. The vertex of the graph is at the point (0, -2). The graph opens upwards, passing through the x-axis at (-2, 0) and (2, 0). The line segment for x < 0 has a slope of -1, and the line segment for x > 0 has a slope of 1. Arrows on both ends indicate that the graph extends indefinitely, representing the function y = |x| - 2.

21.

A graph showing a V-shaped function on a Cartesian coordinate system, with its vertex located at (0, -2).

23.

A graph on a coordinate plane displays a V-shaped function. The vertex of the V is located at the point (0, -2). From this vertex, one arm of the V extends downwards and to the left, passing through approximately (-2, -4), and the other arm extends downwards and to the right, passing through approximately (2, -4). Both arms are indicated with arrows, signifying that the function continues indefinitely in those directions. The coordinate plane shows x-axis values from -5 to 5 and y-axis values from -7 to 5, with grid lines at each integer increment.

25.

A graph showing an absolute value function in a V-shape. The vertex of the function is located at the point (-3, 1), and the graph opens upwards on a coordinate plane.

27.

A graph displays an absolute value function, forming a V-shape on a Cartesian coordinate plane. The x-axis is labeled from -6 to 5, and the y-axis is labeled from -6 to 6. The graph has its vertex at (2, -3) and opens upwards, passing through points such as (0, 1), (1, -1), (3, -1), and (4, 1).

29.

A coordinate plane shows an absolute value function graphed as a V-shape opening downwards. The x-axis ranges from -6 to 5, and the y-axis ranges from -6 to 6. The vertex of the V-shape is located at the point (1, -3). The two linear segments extend downwards and outwards from the vertex, with arrows indicating that the graph continues indefinitely.

31.

A graph on a coordinate plane shows a V-shaped function. The vertex of the V is located at the point (-4, -3). From this vertex, two arms extend upwards, with arrows indicating they continue infinitely.

33.

range: $[- 400, 100]$

35.

37.

There is no solution for $a$ that will keep the function from having a $y$ -intercept. The absolute value function always crosses the $y$ -intercept when $x = 0.$

39.

$| p - 0.08 | \leq 0.015$

41.

$| x - 5.0 | \leq 0.01$

3.7 Section Exercises

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that $y$ -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no $y$ -values repeat and the function is one-to-one.

Yes. For example, $f (x) = \frac{1}{x}$ is its own inverse.

Given a function $y = f (x),$ solve for $x$ in terms of $y .$ Interchange the $x$ and $y .$ Solve the new equation for $y .$ The expression for $y$ is the inverse, $y = f^{- 1} (x) .$

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

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

11.

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

13.

domain of $f (x) : [- 7, \infty); f^{- 1} (x) = \sqrt{x} - 7$

15.

domain of $f (x) : [0, \infty); f^{- 1} (x) = \sqrt{x + 5}$

16.

ⓐ $f (g (x)) = x$ and $g (f (x)) = x .$
ⓑThis tells us that $f$ and $g$ are inverse functions

17.

$f (g (x)) = x, g (f (x)) = x$

19.

one-to-one

21.

one-to-one

23.

not one-to-one

25.

$3$

27.

$2$

29.

Graph of a square root function and its inverse.

31.

$[2, 10]$

33.

$6$

35.

$- 4$

37.

$0$

39.

$1$

41.

$x$	1	4	7	12	16
$f^{- 1} (x)$	3	6	9	13	14

43.

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

Graph of a cubic function and its inverse.

45.

$f^{- 1} (x) = \frac{5}{9} (x - 32) .$ Given the Fahrenheit temperature, $x,$ this formula allows you to calculate the Celsius temperature.

47.

$t (d) = \frac{d}{50},$ $t (180) = \frac{180}{50} .$ The time for the car to travel 180 miles is 3.6 hours.

Review Exercises

function

not a function

$f (- 3) = - 27;$ $f (2) = - 2;$ $f (- a) = - 2 a^{2} - 3 a;$
$- f (a) = 2 a^{2} - 3 a;$ $f (a + h) = - 2 a^{2} + 3 a - 4 a h + 3 h - 2 h^{2}$

one-to-one

function

11.

function

13.

A graph of an upward-opening parabola with its vertex at (0, -2) and passing through approximately (-1.4, 0) and (1.4, 0). The curve extends upwards on both sides.

15.

$2$

17.

$x = - 1.8$ or $or x = 1.8$

19.

$\frac{- 64 + 80 a - 16 a^{2}}{- 1 + a} = - 16 a + 64$

21.

$(- \infty, - 2) \cup (- 2, 6) \cup (6, \infty)$

23.

A piecewise linear graph featuring a jump discontinuity at x = -2. The function approaches -1 from the left with an open circle, and has a value of 1 from the right, continuing downwards.

25.

$31$

27.

increasing $(2, \infty);$ decreasing $(- \infty, 2)$

29.

increasing $(- 3, 1);$ constant $(- \infty, - 3) \cup (1, \infty)$

31.

local minimum $(- 2, - 3);$ local maximum $(1, 3)$

33.

$(- 1.8, 10)$

35.

$(f \circ g) (x) = 17 - 18 x; (g \circ f) (x) = - 7 - 18 x$

37.

$(f \circ g) (x) = \sqrt{\frac{1}{x} + 2}; (g \circ f) (x) = \frac{1}{\sqrt{x + 2}}$

39.

$(f \circ g) (x) = \frac{1 + x}{1 + 4 x}, x \neq 0, x \neq - \frac{1}{4}$

41.

$(f \circ g) (x) = \frac{1}{\sqrt{x}}, x > 0$

43.

sample: $g (x) = \frac{2 x - 1}{3 x + 4}; f (x) = \sqrt{x}$

45.

47.

49.

51.

A V-shaped graph resembling an absolute value function is shown on a Cartesian plane. Its vertex is at (2, -24), and it crosses the x-axis at (-4, 0) and (8, 0).

53.

55.

$f (x) = | x - 3 |$

57.

even

59.

odd

61.

even

63.

$f (x) = \frac{1}{2} | x + 2 | + 1$

65.

$f (x) = - 3 | x - 3 | + 3$

67.

A graph displays a blue V-shaped function, symmetrical around x=3, with its vertex at (3, 0). The two linear segments extend downwards, passing through points (0, -3) and (6, -3).

69.

$f^{- 1} (x) = \frac{x - 9}{10}$

71.

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

73.

The function is one-to-one.

A graph of the reciprocal function y = 1/x, showing a curve in the first and third quadrants of the Cartesian coordinate system. The x-axis and y-axis are labeled with tick marks from -4 to 4. The curve has a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis). In the first quadrant, the curve starts high near the positive y-axis, passes through (1,1) and extends towards the positive x-axis. In the third quadrant, the curve starts low near the negative y-axis, passes through (-1,-1) and extends towards the negative x-axis.

75.

$5$

Practice Test

The relation is a function.

−16

The graph is a parabola and the graph fails the horizontal line test.

$2 a^{2} - a$

$- 2 (a + b) + 1$

11.

$\sqrt{2}$

13.

A graph displays a curve resembling a square root function, starting at approximately (-6, -1) and extending upward and to the right, passing through (-5, 0).

15.

$even$

17.

$odd$

19.

$f^{- 1} (x) = \frac{x + 5}{3}$

21.

$(- \infty, - 1.1) and (1.1, \infty)$

23.

$(1.1, - 0.9)$

25.

$f (2) = 2$

27.

$f (x) = {\begin{matrix} | x | if x \leq 2 \\ 3 if x > 2 \end{matrix}$

29.

$x = 2$

31.

yes

33.

$f^{- 1} (x) = - \frac{x - 11}{2}$