Calculus: One Variable

Garret J. Etgen, Saturnino L. Salas

Chapter 7

The Transcendental Functions - all with Video Answers

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Section 1

One-to-One Functions; Inverses

Problem 1

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=5 x+3$$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 2

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=3 x+5$$

Tyler Moulton

Numerade Educator

Problem 3

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=1-x^{2}$$

Tyler Moulton

Numerade Educator

Problem 4

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x^{5}$$

Tyler Moulton

Numerade Educator

Problem 5

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x^{5}+1$$

Tyler Moulton

Numerade Educator

Problem 6

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x^{2}-3 x+2$$

Tyler Moulton

Numerade Educator

Problem 7

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=1+3 x^{3}$$

Tyler Moulton

Numerade Educator

Problem 8

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x^{3}-1$$

Tyler Moulton

Numerade Educator

Problem 9

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=(1-x)^{3}$$

Tyler Moulton

Numerade Educator

Problem 10

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=(1-x)^{4}$$

Tyler Moulton

Numerade Educator

Problem 11

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=(x+1)^{3}+2$$

Tyler Moulton

Numerade Educator

Problem 12

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=(4 x-1)^{3}$$

Tyler Moulton

Numerade Educator

Problem 13

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x^{3 / 5}$$

Tyler Moulton

Numerade Educator

Problem 14

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=1-(x-2)^{1 / 3}$$

Tyler Moulton

Numerade Educator

Problem 15

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=(2-3 x)^{3}$$

Tyler Moulton

Numerade Educator

Problem 16

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\left(2-3 x^{2}\right)^{3}$$

Tyler Moulton

Numerade Educator

Problem 17

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\sin x, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Tyler Moulton

Numerade Educator

Problem 18

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\cos x, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Tyler Moulton

Numerade Educator

Problem 19

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{1}{x}$$

Tyler Moulton

Numerade Educator

Problem 20

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{1}{1-x}$$

Tyler Moulton

Numerade Educator

Problem 21

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=x+\frac{1}{x}$$

Tyler Moulton

Numerade Educator

Problem 22

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{x}{|x|}$$

Tyler Moulton

Numerade Educator

Problem 23

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{1}{x^{3}+1}$$

Tyler Moulton

Numerade Educator

Problem 24

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{1}{1-x}-1$$

Tyler Moulton

Numerade Educator

Problem 25

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{x+2}{x+1}$$

Tyler Moulton

Numerade Educator

Problem 26

Determine whether or not the function is one to one and, if so, find the inverse. If the function has an inverse, give the domain of the inverse.
$$f(x)=\frac{1}{(x+1)^{2 / 3}}$$

Tyler Moulton

Numerade Educator

Problem 27

What is the relation between a one-to-one function $f$ and the function $\left(f^{-1}\right)^{-1} ?$

Gregory Higby

Numerade Educator

Problem 28

Sketch the graph of the inverse of the function graphed below.
(FIGURE CANNOT COPY)

Gregory Higby

Numerade Educator

Problem 29

Sketch the graph of the inverse of the function graphed below.
(FIGURE CANNOT COPY)

Gregory Higby

Numerade Educator

Problem 30

Sketch the graph of the inverse of the function graphed below.
(FIGURE CANNOT COPY)

Gregory Higby

Numerade Educator

Problem 31

Sketch the graph of the inverse of the function graphed below.
(FIGURE CANNOT COPY)

Gregory Higby

Numerade Educator

Problem 32

(a) Show that the composition of two one-to-one functions, $f$ and $g,$ is one-to-one.
(b) Express $(f \circ g)^{-1}$ in terms of $f^{-1}$ and $g^{-1}$.

Nick Johnson

Numerade Educator

Problem 33

(a) Let $f(x)=\frac{1}{3} x^{3}+x^{2}+k x, k$ a constant. For what values of $k$ is $f$ one-to-one?
(b) Let $g(x)=x^{3}+k x^{2}+x, k$ a constant. For what values of $k$ is $g$ one-to-one?

Gregory Higby

Numerade Educator

Problem 34

(a) Suppose that $f$ has an inverse, $f(2)=5,$ and $f^{\prime}(2)=$ $-\frac{3}{4} .$ What is $\left(f^{-1}\right)^{\prime}(5) ?$
(b) Suppose that $f$ has an inverse, $f(2)=-3,$ and $f^{\prime}(2)=$ $\frac{2}{3} .$ If $g=1 / f^{-1},$ what is $g^{\prime}(-3) ?$

Lauren Shelton

Numerade Educator

Problem 35

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=x^{3}+1 ; \quad c=9$$

Lauren Shelton

Numerade Educator

Problem 36

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=1-2 x-x^{3} ; \quad c=4$$

Gregory Higby

Numerade Educator

Problem 37

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=x+2 \sqrt{x}, \quad x>0 ; \quad c=8$$

Gregory Higby

Numerade Educator

Problem 38

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=\sin x,-\frac{1}{2} \pi<x<\frac{1}{2} \pi ; \quad c=-\frac{1}{2}$$

Gregory Higby

Numerade Educator

Problem 39

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=2 x+\cos x ; \quad c=\pi$$

Gregory Higby

Numerade Educator

Problem 40

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=\frac{x+3}{x-1}, \quad x>1 ; \quad c=3$$

Gregory Higby

Numerade Educator

Problem 41

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=\tan x,-\frac{1}{2} \pi<x<\frac{1}{2} \pi ; \quad c=\sqrt{3}$$

Gregory Higby

Numerade Educator

Problem 42

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=x^{5}+2 x^{3}+2 x ; \quad c=-5$$

Gregory Higby

Numerade Educator

Problem 43

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=3 x-\frac{1}{x^{3}}, \quad x>0 ; \quad c=2$$

Gregory Higby

Numerade Educator

Problem 44

Verify that $f$ has an inverse and find $\left(f^{-1}\right)^{\prime}(c).$
$$f(x)=x-\pi+\cos x, \quad 0<x<2 \pi ; \quad c=-1$$

Gregory Higby

Numerade Educator

Problem 45

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given.
$$f^{\prime}(x)=f(x)$$

Lauren Shelton

Numerade Educator

Problem 46

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given.
$$f^{\prime}(x)=1+[f(x)]^{2}$$

Nick Johnson

Numerade Educator

Problem 47

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given.
$$f^{\prime}(x)=\sqrt{1-[f(x)]^{2}}$$

Nick Johnson

Numerade Educator

Problem 48

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given. Set $$ f(x)=\left\{\begin{array}{ll} x^{3}-1, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. $$
(a) Sketch the graph of $f$ and verify that $f$ is one-to-one.
(b) Find $f^{-1}.$

Gregory Higby

Numerade Educator

Problem 49

For let $f(x)=\frac{a x+b}{c x+d}.$
(a) Show that $f$ is one-to-one iff $a d-b c \neq 0.$
(b) Suppose that $a d-b c \neq 0 .$ Find $f^{-1}.$

Nick Johnson

Numerade Educator

Problem 50

For let $f(x)=\frac{a x+b}{c x+d}.$
Determine the constants $a, b, c, d$ for which $f=f^{-1}.$

Nick Johnson

Numerade Educator

Problem 51

Set $$ f(x)=\int_{2}^{x} \sqrt{1+t^{2}} d t $$
(a) Show that $f$ has an inverse. (b) Find $\left(f^{-1}\right)^{\prime}(0)$

Gregory Higby

Numerade Educator

Problem 52

Set $$ f(x)=\int_{1}^{2 x} \sqrt{16+t^{4}} d t $$
(a) Show that $f$ has an inverse.
(b) Find $\left(f^{-1}\right)^{\prime}(0).$

Gregory Higby

Numerade Educator

Problem 53

Let $f$ be a twice differentiable one-to-one function and set $g=f^{-1}$
(a) Show that
$$ g^{\prime \prime}(x)=-\frac{f^{\prime \prime}[g(x)]}{\left(f^{\prime}[g(x)]\right)^{3}} $$
(b) Suppose that the graph of $f$ is concave up (down). What can you say then about the graph of $f^{-1} ?$

Mengchun Cai

Numerade Educator

Problem 54

Let $P$ be a polynomial of degree $n$
(a) Can $P$ have an inverse if $n$ is even? Support your answer.
(b) Can $P$ have an inverse if $n$ is odd? If so, give an example. Then give an example of a polynomial of odd degree that docs not have an inverse.

Lucas Finney

Numerade Educator

Problem 55

The function $f(x)=\sin x,-\pi / 2<x<\pi / 2,$ is one to one, differentiable, and its derivative does not take on the value 0. Thus $f$ has a differentiable inverse $y=f^{-1}(x)$ Find $d y / d x$ by setting $f(y)=x$ and differentiating implicitly. Express the result as a function of $x$.

Gregory Higby

Numerade Educator

Problem 56

Exercise 55 for $f(x)=\tan x,-\pi / 2<x<\pi / 2.$

Gregory Higby

Numerade Educator

Problem 57

Find $f^{-1}.$
$$f(x)=4+3 \sqrt{x-1}, \quad x \geq 1$$

Adrian Co

Numerade Educator

Problem 58

Find $f^{-1}.$
$$f(x)=\frac{3 x}{2 x+5}, \quad x \neq-5 / 2$$

Adrian Co

Numerade Educator

Problem 59

Find $f^{-1}.$
$$f(x)=\sqrt[3]{8-x}+2$$

Adrian Co

Numerade Educator

Problem 60

Find $f^{-1}.$
$$f(x)=\frac{1-x}{1+x}$$

Adrian Co

Numerade Educator

Problem 61

Use a graphing utility to draw the graph of $f$ Show that $f$ is one-to-one by consideration of $f^{\prime} .$ Draw a figure that displays both the graph of $f$ and the graph of $f^{-1}.,$
$$f(x)=x^{3}+3 x+2$$

Gregory Higby

Numerade Educator

Problem 62

Use a graphing utility to draw the graph of $f$ Show that $f$ is one-to-one by consideration of $f^{\prime} .$ Draw a figure that displays both the graph of $f$ and the graph of $f^{-1}.,$
$$f(x)=x^{3 / 5}-1$$

Gregory Higby

Numerade Educator

Problem 63

Use a graphing utility to draw the graph of $f$ Show that $f$ is one-to-one by consideration of $f^{\prime} .$ Draw a figure that displays both the graph of $f$ and the graph of $f^{-1}.,$
$$f(x)=4 \sin 2 x$$

Gregory Higby

Numerade Educator

Problem 64

Use a graphing utility to draw the graph of $f$ Show that $f$ is one-to-one by consideration of $f^{\prime} .$ Draw a figure that displays both the graph of $f$ and the graph of $f^{-1}.,$
$$f(x)=2-\cos 3 x$$

Gregory Higby

Numerade Educator