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Chapter 1

Functions and Models

Educators

SB

Problem 1

Suppose the graph of $ f $ is given. Write equations for the graphs that are obtained from the graph of
$ f $ as follows.

(a) Shift 3 units upward.
(b) Shift 3 units downward.
(c) Shift 3 units to the right.
(d) Shift 3 units to the left.
(e) Reflect about the x-axis.
(f) Reflect about the y-axis.
(g) Stretch vertically by a factor of 3.
(h) Shrink vertically by a factor of 3.

Heather Z.
Numerade Educator

Problem 2

Explain how each graph is obtained from the graph of $ y = f(x) $.

(a) $ y = f(x) + 8 $
(b) $ y = f (x + 8) $
(c) $ y = 8f(x) $
(d) $ y = f(8x) $
(e) $ y = -f(x) - 1 $
(f) $ y = 8f (\frac{1}{8}x) $

Tyler M.
Numerade Educator

Problem 3

The graph of $ y = f(x) $ is given. Match each equation with its graph and give reasons for your choices.

(a) $ y = f (x - 4) $
(b) $ y = f(x) + 3 $
(c) $ y = \frac{1}{3} f(x) $
(d) $ y = -f (x + 4) $
(e) $ y = 2f (x + 6) $

Heather Z.
Numerade Educator

Problem 4

The graph of $ f $ is given. Draw the graphs of the following functions.

(a) $ y = f(x) - 3 $
(b) $ y = f (x + 1) $
(c) $ y = \frac{1}{2} f(x) $
(d) $ y = -f(x) $

Heather Z.
Numerade Educator

Problem 5

The graph of $ f $ is given. Use it to graph the following functions.

(a) $ y = f (2x) $
(b) $ y = f (\frac{1}{2} x) $
(c) $ y = f (-x) $
(d) $ y = -f (-x) $

Heather Z.
Numerade Educator

Problem 6

The graph of $ y = \sqrt{3x - x^2} $ is given. Use transformations to create a function whose graph is as shown.

Heather Z.
Numerade Educator

Problem 7

The graph of $ y = \sqrt{3x - x^2} $ is given. Use transformations to create a function whose graph is as shown.

Heather Z.
Numerade Educator

Problem 8

(a) How is the graph of $ y = 2 \sin x $ related to the graph of $ y = \sin x $? Use your answer and Figure 6 to sketch the graph of $ y = 2 \sin x $.

(b) How is the graph of $ y = 1 + \sqrt{x} $ related to the graph of $ y = \sqrt {x} $? Use your answer and Figure 4(a) to sketch the graph of $ y = 1 + \sqrt{x} $.

Heather Z.
Numerade Educator

Problem 9

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = - x^2 $

Heather Z.
Numerade Educator

Problem 10

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = (x - 3)^2 $

Heather Z.
Numerade Educator

Problem 11

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = x^3 + 1 $

Heather Z.
Numerade Educator

Problem 12

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 1 - \frac{1}{x} $

Heather Z.
Numerade Educator

Problem 13

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 2 \cos 3x $

Heather Z.
Numerade Educator

Problem 14

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 2 \sqrt{x + 1} $

Heather Z.
Numerade Educator

Problem 15

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = x^2 - 4x + 5 $

Heather Z.
Numerade Educator

Problem 16

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 1 + \sin \pi x $

Heather Z.
Numerade Educator

Problem 17

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 2 - \sqrt{x} $

Heather Z.
Numerade Educator

Problem 18

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 3 - 2 \cos x $

Heather Z.
Numerade Educator

Problem 19

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

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

Heather Z.
Numerade Educator

Problem 20

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = \mid x \mid - 2 $

Heather Z.
Numerade Educator

Problem 21

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = \mid x - 2 \mid $

Heather Z.
Numerade Educator

Problem 22

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = \frac{1}{4} \tan (x - \frac{\pi}{4}) $

Heather Z.
Numerade Educator

Problem 23

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = \mid \sqrt{x} - 1 \mid $

Heather Z.
Numerade Educator

Problem 24

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = \mid \cos \pi x \mid $

Heather Z.
Numerade Educator

Problem 25

The city of New Orleans is located at latitude 30 $ ^{\circ} N $. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 am and sets at 6:18 pm in
New Orleans.

Yiming Z.
Numerade Educator

Problem 26

A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by $ \pm 0.35 $ magnitude. Find
a function that models the brightness of Delta Cephei as a function of time.

Clarissa N.
Numerade Educator

Problem 27

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth $ D(t) $ (in meters) as a function
of time $ t $ (in hours after midnight) on that day.

Linda H.
Numerade Educator

Problem 28

In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air
$ V(t) $ in the lungs as a function of time.

Jeffrey P.
Numerade Educator

Problem 29

(a) How is the graph of $ y = f (\mid x \mid) $ related to the graph of $ f $.

(b) Sketch the graph of $ y = \sin \mid x \mid $.

(c) Sketch the graph of $ y = \sqrt{\mid x \mid} $.

Heather Z.
Numerade Educator

Problem 30

Use the given graph of $ f $ to sketch the graph of $ y = \frac{1}{f(x)} $. Which features of $ f $ are the most important in sketching $ y = \frac{1}{f(x)} $? Explain how they are used.

Heather Z.
Numerade Educator

Problem 31

Find (a) $ f + g $, (b) $ f - g $, (c) $ fg $ and (d) $ f/g $ and state their domains.

$ f(x) = x^3 + 2x^2 $ , $ g(x) = 3x^2 - 1 $

Heather Z.
Numerade Educator

Problem 32

Find (a) $ f + g $, (b) $ f - g $, (c) $ fg $ and (d) $ f/g $ and state their domains.

$ f(x) = \sqrt {3 - x} $ , $ g(x) = \sqrt {x^2 -1} $

Heather Z.
Numerade Educator

Problem 33

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = 3x + 5 $ , $ g(x) = x^2 + x $

Heather Z.
Numerade Educator

Problem 34

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = x^3 - 2 $ , $ g(x) = 1 - 4x $

Heather Z.
Numerade Educator

Problem 35

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = \sqrt {x + 1} $ , $ g(x) = 4x - 3 $

Heather Z.
Numerade Educator

Problem 36

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = \sin x $ , $ g(x) = x^2 + 1 $

Heather Z.
Numerade Educator

Problem 37

Find the functions (a) $f \circ g,$ (b) $g \circ f,$ (c) $f \circ f,$ and (d) $g \circ g$ and their domains.
$$
f(x)=x+\frac{1}{x}, \quad g(x)=\frac{x+1}{x+2}
$$

Heather Z.
Numerade Educator

Problem 38

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = \dfrac{x}{1 + x} $ , $ g(x) = \sin 2x $

Heather Z.
Numerade Educator

Problem 39

Find $ f \circ g \circ h $.

$ f(x) = 3x -2 $ , $ g(x) = \sin x $ , $ h(x) = x^2 $

Heather Z.
Numerade Educator

Problem 40

Find $ f \circ g \circ h $.

$ f(x) = \mid x - 4 \mid $ , $ g(x) = 2^x $ , $ h(x) = \sqrt{x} $

Heather Z.
Numerade Educator

Problem 41

Find $ f \circ g \circ h $.

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

Heather Z.
Numerade Educator

Problem 42

Find $ f \circ g \circ h $.

$ f(x) = \tan x $ , $ g(x) = \dfrac{x}{x - 1} $ , $ h(x) = \sqrt[3]{x} $

Heather Z.
Numerade Educator

Problem 43

Express the function in the form $ f \circ g $.

$ F(x) = (2x + x^2)^4 $

Heather Z.
Numerade Educator

Problem 44

Express the function in the form $ f \circ g $.

$ F(x) = \cos^2 x $

Heather Z.
Numerade Educator

Problem 45

Express the function in the form $ f \circ g $.

$ F(x) = \dfrac{\sqrt[3]{x}}{1 + \sqrt[3]{x}} $

Heather Z.
Numerade Educator

Problem 46

Express the function in the form $ f \circ g $.

$ G(x) = \sqrt[3]{\dfrac {x}{1 + x}} $

Heather Z.
Numerade Educator

Problem 47

Express the function in the form $ f \circ g $.

$ v(t) = \sec (t^2) \tan (t^2) $

Heather Z.
Numerade Educator

Problem 48

Express the function in the form $ f \circ g $.

$ u(t) = \dfrac {\tan t}{1 + \tan t} $

Heather Z.
Numerade Educator

Problem 49

Express the function in the form $ f \circ g \circ h $.

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

Heather Z.
Numerade Educator

Problem 50

Express the function in the form $ f \circ g \circ h $.

$ H(x) = \sqrt[8]{2 + \mid x \mid} $

Heather Z.
Numerade Educator

Problem 51

Express the function in the form $ f \circ g \circ h $.

$ S(t) = \sin^2 (\cos t) $

Heather Z.
Numerade Educator

Problem 52

Use the table to evaluate each expression.

(a) $ f (g(1)) $
(b) $ g (f(1)) $
(c) $ f (f(1)) $
(d) $ g (g(1)) $
(e) $ (g \circ f) (3) $
(f) $ (f \circ g) (6) $

Heather Z.
Numerade Educator

Problem 53

Use the given graphs of $ f $ and $ g $ to evaluate each expression, or explain why it is undefined.

(a) $ f (g(2)) $
(b) $ g (f(0)) $
(c) $ (f \circ g) (0) $
(d) $ (g \circ f) (6) $
(e) $ (g \circ g) (-2) $
(f) $ (f \circ f) (4) $

Heather Z.
Numerade Educator

Problem 54

Use the given graphs of $ f $ and $ g $ to estimate the value of $ f (g(x)) $ for $ x = -5, -4, -3, . . . , 5 $. Use these estimates to sketch a rough graph of $ f \circ g $.

Heather Z.
Numerade Educator

Problem 55

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

(a) Express the radius $ r $ of this circle as a function of the time $ t $ (in seconds).
(b) If $ A $ is the area of this circle as a function of the radius, find $ A \circ r $ and interpret it.

Heather Z.
Numerade Educator

Problem 56

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s.

(a) Express the radius $ r $ of the balloon as a function of the time $ t $ (in seconds).
(b) If $ V $ is the volume of the balloon as a function of the radius, find $ V \circ r $ and interpret it.

am
Abhidipta M.
Numerade Educator

Problem 57

A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon.

(a) Express the distance $ s $ between the lighthouse and the ship as a function of $ d $, the distance the ship has traveled since noon; that is, find $ f $ so that $ s = f(d) $.
(b) Express $ d $ as a function of $ t $, the time elapsed since noon; that is, find $ t $ so that $ d = g(t) $.
(c) Find $ f \circ g $. What does this function represent?

SD
Shane D.
Numerade Educator

Problem 58

An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time $ t = 0 $.

(a) Express the horizontal distance $ d $ (in miles) that the plane has flown as a function of $ t $ .
(b) Express the distance $ s $ between the plane and the radar station as a function of $ d $ .
(c) Use composition to express s as a function of $ t $.

Heather Z.
Numerade Educator

Problem 59

The Heaviside function $ H $ is defined by
$ H(t) = \left\{
\begin{array}{ll}
0 & \mbox{if $ t < 0 $}\\
1 & \mbox{if $ t \ge 0 $}
\end{array} \right.$
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on.
(a) Sketch the graph of the Heaviside function.
(b) Sketch the graph of the voltage $ V(t) $ in a circuit if the switch is turned on at time $ t = 0 $ and 120 volts are applied instantaneously to the circuit. Write a formula for $ V(t) $ in terms of $ H(t) $.
(c) Sketch the graph of the voltage $ V(t) $ in a circuit if the switch is turned on at time $ t = 5 $ seconds and 240 volts are applied instantaneously to the circuit. Write a formula for $ V(t) $ in terms of $ H(t) $. (Note that starting at $ t = 5 $ corresponds to a translation.)

Jeffrey P.
Numerade Educator

Problem 60

The Heaviside function defined in Exercise 59 can also be used to define the ramp function $ y = ctH(t) $ , which represents a gradual increase in voltage or current in a circuit.

(a) Sketch the graph of the ramp function $ y = tH(t) $.
(b) Sketch the graph of the voltage $ V(t) $ in a circuit if the switch is turned on at time $ t = 0 $ and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for $ V(t) $ in terms of $ H(t) $ for $ t \le 60 $.
(c) Sketch the graph of the voltage $ V(t) $ in a circuit if the switch is turned on at time $ t = 7 $ seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for $ V(t) $ in terms of $ H(t) $ for $ t \le 32 $.

Carson M.
Numerade Educator

Problem 61

Let $ f $ and $ g $ be linear functions with equations $ f(x) = m_1 x + b_1 $ and and $ g(x) = m_2 x + b_2 $. Is $ f \circ g $ also a linear function? If so, what is the slope of its graph?

Heather Z.
Numerade Educator

Problem 62

If you invest $ x $ dollars at 4% interest compounded annually, then the amount $ A(x) $ of the investment after one year is $ A(x) = 1.04x $. Find $ A \circ A $, $ A \circ A \circ A $, and $ A \circ A \circ A \circ A $. What do these compositions represent? Find a formula for the composition of $ n $ copies of $ A $.

Heather Z.
Numerade Educator

Problem 63

(a) If $ g(x) = 2x + 1 $ and $ h(x) = 4x^2 + 4x + 7 $, find a function $ f $ such that $ f \circ g = h $. (Think about what operations you would have to perform on the formula for $ g $ to end up with the formula for $ h $.)
(b) If $ f(x) = 3x + 5 $ and $ h(x) = 3x^2 + 3x + 2 $, find a function $ g $ such that $ f \circ g = h $.

Heather Z.
Numerade Educator

Problem 64

If $ f(x) = x + 4 $ and $ h(x) = 4x - 1 $, find a function $ g $ such that $ g \circ f = h $.

Heather Z.
Numerade Educator

Problem 65

Suppose $ g $ is an even function and let $ h = f \circ g $. Is $ h $ always an even function?

Jeffrey P.
Numerade Educator

Problem 66

Suppose $ g $ is an odd function and let $ h = f \circ g $. Is $ h $ always an odd function? What if $ f $ is odd? What if $ f $ is even?

Jeffrey P.
Numerade Educator