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Chapter 3

INVERSE FUNCTIONS

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Problem 1

(a) Write an equation that defines the exponential function
with base $a>0$ .
(b) What is the domain of this function?
(c) If $a \neq 1,$ what is the range of this function?
(d) Sketch the general shape of the graph of the exponential function for each of the following cases.
(i) $a>1$

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Problem 2

(a) How is the number $e$ defined?
(b) What is an approximate value for $e ?$
(c) What is the natural exponential function?

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Problem 3

Graph the given functions on a common screen. How are these graphs related?
$$y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$$

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Problem 4

Graph the given functions on a common screen. How are these graphs related?
$$y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$$

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Problem 5

Graph the given functions on a common screen. How are these graphs related?
$$y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$$

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Problem 6

Graph the given functions on a common screen. How are these graphs related?
$$y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$$

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Problem 7

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$y=10^{x+2}$

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Problem 8

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$$y=(0.5)^{x}-2$$

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Problem 9

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$$y=-2^{-x}$$

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Problem 10

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$$y=e^{|x|}$$

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Problem 11

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$$y=1-\frac{1}{2} e^{-x}$$

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Problem 12

Make a rough sketch of the graph of the function. Do
not use a calculator. Just use the graphs given in Figures 3 and
9 and, if necessary, the transformations of Section 1.2 .
$$y=2\left(1-e^{x}\right)$$

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Problem 13

Starting with the graph of $y=e^{x},$ write the equation of the
graph that results from
(a) shifting 2 units downward
(b) shifting 2 units to the right
(c) reflecting about the $x$ -axis
(d) reflecting about the $y$ -axis
(e) reflecting about the $x$ -axis and then about the $y$ -axis

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Problem 14

Starting with the graph of $y=e^{x},$ find the equation of the
graph that results from
(a) reflecting about the line $y=4$
(b) reflecting about the line $x=2$

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Problem 15

Find the domain of each function.
$$f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$$

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Problem 15

Find the domain of each function.
$$f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$$ $$f(x)=\frac{1+x}{e^{\cos x}}$$

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Problem 16

Find the domain of each function.
$$g(t)=\sin \left(e^{-t}\right) \quad \text { (b) } g(t)=\sqrt{1-2^{t}}$$

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Problem 17

Find the exponential function $f(x)=C a^{x}$ whose graph is given.

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Problem 18

Find the exponential function $f(x)=C a^{x}$ whose graph is given.

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Problem 19

Suppose the graphs of $f(x)=x^{2}$ and $g(x)=2^{x}$ are drawn
on a coordinate grid where the unit of measurement is
1 inch. Show that, at a distance 2 ft to the right of the ori-
gin, the height of the graph of $f$ is 48 ft but the height
of the graph of $g$ is about 265 mi.

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Problem 20

Compare the rates of growth of the functions $f(x)=x^{5}$
and $g(x)=5^{x}$ by graphing both functions in several view-
ing rectangles. Find all points of intersection of the graphs
correct to one decimal place.

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Problem 21

Compare the functions $f(x)=x^{10}$ and $g(x)=e^{x}$ by graph-
ing both $f$ and $g$ in several viewing rectangles. When does
the graph of $g$ finally surpass the graph of $f ?$

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Problem 22

Use a graph to estimate the values of $x$ such that
$e^{x}>1,000,000,000 .$

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Problem 23

Find the limit.
$$\lim _{x \rightarrow \infty}(1.001)^{x}$$

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Problem 24

Find the limit.
$$\lim _{x \rightarrow \infty} e^{-x^{2}}$$

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Problem 25

Find the limit.
$$\lim _{x \rightarrow \infty} \frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}}$$

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Problem 26

Find the limit.
$$\lim _{x \rightarrow \infty} \frac{2+10^{x}}{3-10^{x}}$$

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Problem 27

Find the limit.
$$\lim _{x \rightarrow 2^{+}} e^{3 /(2-x)}$$

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Problem 28

Find the limit.
$$\lim _{x \rightarrow 2^{-}} e^{3 /(2-x)}$$

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Problem 29

Find the limit.
$$\lim _{x \rightarrow \infty}\left(e^{-2 x} \cos x\right)$$

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Problem 30

Find the limit.
$$\lim _{x \rightarrow(\pi / 2)^{+}} e^{\tan x}$$

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Problem 31

If you graph the function
$f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}$
you'll see that $f$ appears to be an odd function. Prove it.

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Problem 32

Graph several members of the family of functions
$f(x)=\frac{1}{1+a e^{b x}}$
where $a>0 .$ How does the graph change when $b$
changes? How does it change when $a$ changes?

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