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Chapter 1

Functions and Models

Educators

SB

Problem 1

(a) What is a one-to-one function?
(b) How can you tell from the graph of a function whether it is one-to-one?

Heather Z.
Numerade Educator

Problem 2

(a) Suppose $ f $ is a one-to-one function with domain $ A $ and range $ B $. How is the inverse function $ f^{-1} $ defined? What is the domain of $ f^{-1} $? What is the range of $ f^{-1} $?
(b) If you are given a formula for $ f $, how do you find a formula for $ f^{-1} $?
(c) If you are given the graph of $ f $, how do you find the graph of $ f^{-1} $?

Heather Z.
Numerade Educator

Problem 3

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 4

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 5

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 6

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 7

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 8

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

Heather Z.
Numerade Educator

Problem 9

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ f(x) = 2x - 3 $

Heather Z.
Numerade Educator

Problem 10

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ f(x) = x^4 - 16 $

Heather Z.
Numerade Educator

Problem 11

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ g(x) = 1 - \sin x $

Heather Z.
Numerade Educator

Problem 12

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ g(x) = \sqrt[3]{x} $

Heather Z.
Numerade Educator

Problem 13

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ f(t) $ is the height of a football $ t $ seconds after kickoff.

Heather Z.
Numerade Educator

Problem 14

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

$ f(t) $ is your height at age $ t $.

Heather Z.
Numerade Educator

Problem 15

Assume that $ f $ is a one-to-one function.

(a) If $ f(6) = 17 $ , what is $ f^{-1} (17) $?
(a) If $ f^{-1} (3) = 2 $ , what is $ f(2) $?

Heather Z.
Numerade Educator

Problem 16

If $ f(x) = x^5 + x^3 + x $ , find $ f^{-1} (3) $ and $ f (f^{-1} (2)) $.

Heather Z.
Numerade Educator

Problem 17

If $ g(x) = 3 + x + e^x $ , find $ g^{-1} (4) $.

Heather Z.
Numerade Educator

Problem 18

The graph of $ f $ is given.
(a) Why is $ f $ one-to-one?
(b) What are the domain and range of $ f^{-1} $ ?
(c) What is the value of $ f^{-1} (2) $?
(d) Estimate the value of $ f^{-1} (0) $.

Heather Z.
Numerade Educator

Problem 19

The formula $ C = \frac{5}{9} (F - 32) $ , where $ F \ge -459.67 $, expresses the Celsius temperature $ C $ as a function of the Fahrenheit temperature $ F $. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

Heather Z.
Numerade Educator

Problem 20

In the theory of relativity, the mass of a particle with speed $ v $ is

$$ m = f(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} $$

where $ m_0 $ is the rest mass of the particle and $ c $ is the speed of light in a vacuum. Find the inverse function of $ f $ and explain its meaning.

Heather Z.
Numerade Educator

Problem 21

Find a formula for the inverse of the function.

$ f(x) = 1 + \sqrt{2 + 3x} $

Heather Z.
Numerade Educator

Problem 22

Find a formula for the inverse of the function.

$ f(x) = \dfrac{4x - 1}{2x + 3} $

Heather Z.
Numerade Educator

Problem 23

Find a formula for the inverse of the function.

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

Heather Z.
Numerade Educator

Problem 24

Find a formula for the inverse of the function.

$ y = x^2 - x $ , $ x \ge \frac{1}{2} $

Heather Z.
Numerade Educator

Problem 25

Find a formula for the inverse of the function.

$ y = \ln (x + 3) $

Heather Z.
Numerade Educator

Problem 26

Find a formula for the inverse of the function.

$ y = \dfrac{1 - e^{-x}}{1 + e^{-x}} $

Heather Z.
Numerade Educator

Problem 27

Find an explicit formula for $ f^{-1} $ and use it to graph $ f^{-1} $, $ f $, and the line $ y = x $ on the same screen. To check your work, see whether the graphs of $ f $ and $ f^{-1} $ are reflections about the line.

$ f(x) = \sqrt{4x +3} $

Heather Z.
Numerade Educator

Problem 28

Find an explicit formula for $ f^{-1} $ and use it to graph $ f^{-1} $, $ f $, and the line $ y = x $ on the same screen. To check your work, see whether the graphs of $ f $ and $ f^{-1} $ are reflections about the line.

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

Heather Z.
Numerade Educator

Problem 29

Use the given graph of $ f $ to sketch the graph of $ f^{-1} $.

Heather Z.
Numerade Educator

Problem 30

Use the given graph of $ f $ to sketch the graph of $ f^{-1} $.

Heather Z.
Numerade Educator

Problem 31

Let $ f(x) = \sqrt{1 - x^2} $ , $ 0 \le x \le 1 $.

(a) Find $ f^{-1} $. How is it related to $ f $ ?
(b) Identify the graph of $ f $ and explain your answer to part (a).

Heather Z.
Numerade Educator

Problem 32

Let $ g(x) = \sqrt[3]{1 - x^3} $.

(a) Find $ g^{-1} $ . How is it related to $ g $?
(b) Graph $ g $. How do you explain your answer to part (a)?

Heather Z.
Numerade Educator

Problem 33

(a) How is the logarithmic function $ y = \log_b x $ defined?
(b) What is the domain of this function?
(c) What is the range of this function?
(d) Sketch the general shape of the graph of the function $ y = \log_b x $ if $ b > 1 $.

Heather Z.
Numerade Educator

Problem 34

(a) What is the natural logarithm?
(b) What is the common logarithm?
(c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.

Heather Z.
Numerade Educator

Problem 35

Find the exact value of each expression.

(a) $ \log_2 32 $
(b) $ \log_8 2 $

Heather Z.
Numerade Educator

Problem 36

Find the exact value of each expression.

(a) $ \log_5 \frac{1}{125} $
(b) $ \ln (\frac{1}{e^2}) $

Heather Z.
Numerade Educator

Problem 37

Find the exact value of each expression.

(a) $ \log_{10} 40 + \log_{10} 2.5 $
(b) $ \log_8 60 - \log_8 3 - \log_8 5 $

Heather Z.
Numerade Educator

Problem 38

Find the exact value of each expression.

(a) $ e^{-ln 2} $
(b) $ e^{\ln}^{(\ln e^3)} $

Heather Z.
Numerade Educator

Problem 39

Express the given quantity as a single logarithm.

$ \ln 10 + 2 \ln 5 $

Heather Z.
Numerade Educator

Problem 40

Express the given quantity as a single logarithm.

$ \ln b + 2 \ln c - 3 \ln d $

Heather Z.
Numerade Educator

Problem 41

Express the given quantity as a single logarithm.

$ \frac{1}{3} \ln (x + 2)^3 + \frac{1}{2} [\ln x - \ln (x^2 + 3x + 2)^2] $

Heather Z.
Numerade Educator

Problem 42

Use Formula 10 to evaluate each logarithm correct to six decimal places.

(a) $ \log_5 10 $
(b) $ \log_3 57 $

Heather Z.
Numerade Educator

Problem 43

Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

$ y = \log_{1.5} x $ , $ y = \ln x $ , $ y = \log_{10} x $ , $ y = \log_{50} x $

Heather Z.
Numerade Educator

Problem 44

Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

$ y = \ln x $ , $ y = \log_{10} x $ , $ y = e^x $ , $ y = 10^x $

Heather Z.
Numerade Educator

Problem 45

Suppose that the graph of $ y = \log_2 x $ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?

Heather Z.
Numerade Educator

Problem 46

Compare the functions $ f(x) = x^{0.1} $ and $ g(x) = \ln x $ by graphing both $ f $ and $ g $ in several viewing rectangles. When does the graph of $ f $ finally surpass the graph of $ g $?

Heather Z.
Numerade Educator

Problem 47

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

(a) $ y = \log_{10} (x + 5) $
(b) $ y = -\ln x $

Heather Z.
Numerade Educator

Problem 48

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

(a) $ y = \ln (-x) $
(b) $ y = \ln \mid x \mid $

Heather Z.
Numerade Educator

Problem 49

(a) What are the domain and range of $ f $?
(b) What is the x-intercept of the graph of $ f $?
(c) Sketch the graph of $ f $.

$ f(x) = \ln x + 2 $

Heather Z.
Numerade Educator

Problem 50

(a) What are the domain and range of $ f $?
(b) What is the x-intercept of the graph of $ f $?
(c) Sketch the graph of $ f $.

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

Heather Z.
Numerade Educator

Problem 51

Solve each equation for x.

(a) $ e^{7 - 4x} = 6 $
(b) $ \ln (3x - 10) = 2 $

Heather Z.
Numerade Educator

Problem 52

Solve each equation for x.

(a) $ \ln (x^2 - 1) = 3 $
(b) $ e^{2x} - 3e^x + 2 = 0 $

Heather Z.
Numerade Educator

Problem 53

Solve each equation for x.

(a) $ 2^{x - 5} = 3 $
(b) $ \ln x + \ln (x - 1) = 1 $

Heather Z.
Numerade Educator

Problem 54

Solve each equation for x.

(a) $ \ln (\ln x) = 1 $
(b) $ e^{ax} = Ce^{bx} $ , where $ a \neq b $

Heather Z.
Numerade Educator

Problem 55

Solve each inequality for x.

(a) $ \ln x < 0 $
(b) $ e^x > 5 $

Heather Z.
Numerade Educator

Problem 56

Solve each inequality for x.

(a) $ 1< e^{3x - 1} < 2 $
(b) $ 1 - 2 \ln x < 3 $

Heather Z.
Numerade Educator

Problem 57

(a) We must have $e^{x}-3>0 \Leftrightarrow e^{x}>3 \Leftrightarrow x>\ln 3$. Thus, the domain of $f(x)=\ln \left(e^{x}-3\right)$ is $(\ln 3, \infty)$.
(b) $y=\ln \left(e^{x}-3\right) \Rightarrow e^{y}=e^{x}-3 \Rightarrow e^{x}=e^{y}+3 \Rightarrow x=\ln \left(e^{y}+3\right),$ so $f^{-1}(x)=\ln \left(e^{x}+3\right)$.
Now $e^{x}+3>0 \Rightarrow e^{x}>-3,$ which is true for any real $x,$ so the domain of $f^{-1}$ is $\mathbb{R}$.

Heather Z.
Numerade Educator

Problem 58

(a) What are the values of $ e^{\ln 300} $ and $ \ln (e^{300}) $?
(b) Use your calculator to evaluate $ e^{\ln 300} $ and $ \ln (e^{300}) $. What do you notice? Can you explain why the calculator has trouble?

Heather Z.
Numerade Educator

Problem 59

Graph the function $ f(x) = \sqrt{x^3 + x^2 + x +1} $ and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for $ f^{-1} (x) $. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

Clarissa N.
Numerade Educator

Problem 60

(a) If $ g(x) = x^6 + x^4 $ , $ x \ge 0 $, use a computer algebra system to find an expression for
$ g^{-1} (x) $.
(b) Use the expression in part (a) to graph $ y = g(x) $ , $ y = x $ , and $ y = g^{-1} (x) $ on the same screen.

Clarissa N.
Numerade Educator

Problem 61

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after $ t $ hours is $ n = f(t) = 100 \cdot 2^\frac{t}{3} $.

(a) Find the inverse of this function and explain its meaning.
(b) When will the population reach 50,000?

Heather Z.
Numerade Educator

Problem 62

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by
$$ Q(t) = Q_0 (1 - e^{\frac{-t}{a}}) $$
(The maximum charge capacity is $ Q_0 $ and $ t $ is measured in seconds.)
(a) Find the inverse of this function and explain its meaning.
(b) How long does it take to recharge the capacitor to 90% of capacity if $ a = 2 $?

Heather Z.
Numerade Educator

Problem 63

Find the exact value of each expression.

(a) $ \cos^{-1} (-1) $
(b) $ \sin^{-1} (0.5) $

Heather Z.
Numerade Educator

Problem 64

Find the exact value of each expression.

(a) $ \tan^{-1} \sqrt{3} $
(b) $ \arctan (-1) $

Heather Z.
Numerade Educator

Problem 65

Find the exact value of each expression.

(a) $ \csc^{-1} \sqrt{2} $
(b) $ \arcsin 1 $

Heather Z.
Numerade Educator

Problem 66

Find the exact value of each expression.

(a) $ \sin^{-1} (\frac{-1}{\sqrt{2}}) $
(b) $ \cos^{-1} (\frac{\sqrt{3}}{2}) $

Heather Z.
Numerade Educator

Problem 67

Find the exact value of each expression.

(a) $ \cot^{-1} (-\sqrt{3}) $
(b) $ \sec^{-1} 2 $

Heather Z.
Numerade Educator

Problem 68

Find the exact value of each expression.

(a) $ \arcsin (\sin (\frac{5\pi}{4})) $
(b) $ \cos (2 \sin^{-1} (\frac{5}{13})) $

Heather Z.
Numerade Educator

Problem 69

Prove that $ \cos (\sin^{-1} x) = \sqrt{1 - x^2} $.

Heather Z.
Numerade Educator

Problem 70

Simplify the expression.

$ \tan (\sin^{-1} x) $

Heather Z.
Numerade Educator

Problem 71

Simplify the expression.

$ \sin(\tan^{-1} x) $

Heather Z.
Numerade Educator

Problem 72

Simplify the expression.

$ \sin (2 \arccos x) $

Heather Z.
Numerade Educator

Problem 73

Graph the given functions on the same screen. How are these graphs related?

$ y = \sin x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $ ; $ \sin^{-1} x $ ; $ y = x $

Heather Z.
Numerade Educator

Problem 74

Graph the given functions on the same screen. How are these graphs related?

$ y = \tan x $ , $ \frac{-\pi}{2} < x < \frac{\pi}{2} $ ; $ \tan^{-1} x $ ; $ y = x $

Heather Z.
Numerade Educator

Problem 75

Find the domain and range of the function
$$ g(x) = \sin^{-1} (3x + 1) $$

Heather Z.
Numerade Educator

Problem 76

(a) Graph the function $ f(x) = \sin (\sin^{-1} x) $ and explain the appearance of the graph.
(b) Graph the function $ g(x) = \sin^{-1} (\sin x) $. How do you explain the appearance of this graph?

Jeffrey P.
Numerade Educator

Problem 77

(a) If we shift a curve to the left, what happens to its reflection about the line $ y = x $? In view of this geometric principle, find an expression for the inverse of $ g(x) = f (x + c) $, where $ f $ is a one-to-one function.
(b) Find an expression for the inverse of $ h(x) = f (cx) $ , where $ c \neq 0 $.

Heather Z.
Numerade Educator