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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 4

Exponential and Logarithmic Functions

Educators


Problem 1

Determine whether or not the function determined by the given equation is one-to-one.
$$f(x)=2 x-3$$

Bobby B.
University of North Texas

Problem 2

Determine whether or not the function determined by the given equation is one-to-one.
$$G(x)=-3 x+7$$

Bobby B.
University of North Texas

Problem 3

Determine whether or not the function determined by the given equation is one-to-one.
$$h(x)=2 x+5$$

Bobby B.
University of North Texas

Problem 4

Determine whether or not the function determined by the given equation is one-to-one.
$$w(x)=-7 x+9$$

Bobby B.
University of North Texas

Problem 5

Determine whether or not the function determined by the given equation is one-to-one.
$$f(x)=2 x^{2}-7$$

Bobby B.
University of North Texas

Problem 6

Determine whether or not the function determined by the given equation is one-to-one.
$$g(x)=-4 x^{2}+1$$

Bobby B.
University of North Texas

Problem 7

Determine whether or not the function determined by the given equation is one-to-one.
$$h(x)=\sqrt{2 x+3}$$

Bobby B.
University of North Texas

Problem 8

Determine whether or not the function determined by the given equation is one-to-one.
$$r(x)=\sqrt{2-5 x}$$

Bobby B.
University of North Texas

Problem 9

Determine whether or not the function determined by the given equation is one-to-one.
$$v(x)=4 x^{2}+1, x \geq 0$$

Bobby B.
University of North Texas

Problem 10

Determine whether or not the function determined by the given equation is one-to-one.
$$s(x)=-2 x^{2}+6, x \leq 0$$

Bobby B.
University of North Texas

Problem 11

Determine if the equation describes an increasing or decreasing function or neither.
$$f(x)=2 x-3$$

Bobby B.
University of North Texas

Problem 12

Determine if the equation describes an increasing or decreasing function or neither.
$$G(x)=-3 x+7$$

Bobby B.
University of North Texas

Problem 13

Determine if the equation describes an increasing or decreasing function or neither.
$$h(x)=2 x+5$$

Bobby B.
University of North Texas

Problem 14

Determine if the equation describes an increasing or decreasing function or neither.
$$w(x)=-7 x+9$$

Bobby B.
University of North Texas

Problem 15

Determine if the equation describes an increasing or decreasing function or neither.
$$f(x)=2 x^{2}-7$$

Bobby B.
University of North Texas

Problem 16

Determine if the equation describes an increasing or decreasing function or neither.
$$g(x)=-4 x^{2}+1$$

Bobby B.
University of North Texas

Problem 17

Determine if the equation describes an increasing or decreasing function or neither.
$$h(x)=\sqrt{2 x+3}$$

Bobby B.
University of North Texas

Problem 18

Determine if the equation describes an increasing or decreasing function or neither.
$$r(x)=\sqrt{2-5 x}$$

Bobby B.
University of North Texas

Problem 19

Determine if the equation describes an increasing or decreasing function or neither.
$$v(x)=4 x^{2}+1, x \geq 0$$

Bobby B.
University of North Texas

Problem 20

Determine if the equation describes an increasing or decreasing function or neither.
$$s(x)=-2 x^{2}+6, x \leq 0$$

Bobby B.
University of North Texas

Problem 21

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 22

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 23

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 24

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 25

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 26

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 27

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 28

Determine if the given graph represents a one-to-one function.

Bobby B.
University of North Texas

Problem 29

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=2 x-3$$

Bobby B.
University of North Texas

Problem 30

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=-3 x+7$$

Bobby B.
University of North Texas

Problem 31

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=\sqrt{2 x-8}$$

Bobby B.
University of North Texas

Problem 32

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=\sqrt{6-2 x}$$

Bobby B.
University of North Texas

Problem 33

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=x^{2}, x \geq 0$$

Bobby B.
University of North Texas

Problem 34

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=2 x^{2}+1, x \geq 0$$

Bobby B.
University of North Texas

Problem 35

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=-4 x^{2}+7, x \leq 0$$

Bobby B.
University of North Texas

Problem 36

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=8 x^{3}$$

Bobby B.
University of North Texas

Problem 37

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=2 \sqrt[3]{x}$$

Bobby B.
University of North Texas

Problem 38

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.
$$f(x)=(x-1)^{3}$$

Bobby B.
University of North Texas

Problem 39

If $f(x)=2 x-5,$ find $(a) f^{-1}(1),(b) f^{-1}(3)$

Bobby B.
University of North Texas

Problem 40

If $f(x)=-3 x+9,$ find $\left(\text { a) } f^{-1}(2), \text { (b) } f^{-1}(5)\right.$

Bobby B.
University of North Texas

Problem 41

If $f(x)=\sqrt{2 x-1},$ find $(a) f^{-1}(1)$,
(b) $f^{-1}(3)$

Bobby B.
University of North Texas

Problem 42

If $f(x)=\sqrt{3 x+2},$ find (a) $f^{-1}(3),$ (b) $f^{-1}(4)$

Bobby B.
University of North Texas

Problem 43

If $f(x)=\frac{2 x+1}{3 x+2},$ find $\left(\text { a) } f^{-1}(0), \text { (b) } f^{-1}(-1)\right.$

Bobby B.
University of North Texas

Problem 44

If $f(x)=\frac{2 x-3}{3 x+4},$ find (a) $f^{-1}(-3),$ (b) $f^{-1}(3)$

Bobby B.
University of North Texas

Problem 45

Determine the equation of the inverse function.
$$f(x)=5 x-9$$

Bobby B.
University of North Texas

Problem 46

Determine the equation of the inverse function.
$$f(x)=2 x+1$$

Bobby B.
University of North Texas

Problem 47

Determine the equation of the inverse function.
$$f(x)=2 x^{2}+3, x \geq 0$$

Bobby B.
University of North Texas

Problem 48

Determine the equation of the inverse function.
$$f(x)=-5 x^{2}+3, x \leq 0$$

Bobby B.
University of North Texas

Problem 49

Determine the equation of the inverse function.
$$f(x)=\sqrt{2 x+3}$$

Bobby B.
University of North Texas

Problem 50

Determine the equation of the inverse function.
$$f(x)=-\sqrt{5-4 x}$$

Bobby B.
University of North Texas

Problem 51

Determine the equation of the inverse function.
$$f(x)=\frac{2 x+7}{9 x-3}$$

Bobby B.
University of North Texas

Problem 52

Determine the equation of the inverse function.
$$f(x)=\frac{11-3 x}{2 x+5}$$

Bobby B.
University of North Texas

Problem 53

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=2 x+5, g(x)=\frac{x-5}{2}$$

Bobby B.
University of North Texas

Problem 54

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=3 x-7, g(x)=\frac{x+7}{3}$$

Bobby B.
University of North Texas

Problem 55

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=5 x+9, g(x)=\frac{x-9}{5}$$

Bobby B.
University of North Texas

Problem 56

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=2 x-3, g(x)=\frac{x+3}{2}$$

Bobby B.
University of North Texas

Problem 57

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=\sqrt{9-x^{2}}, g(x)=\sqrt{9-x^{2}}$$

Bobby B.
University of North Texas

Problem 58

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=\sqrt{5-x^{2}}, g(x)=\sqrt{5-x^{2}}$$

Bobby B.
University of North Texas

Problem 59

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=\sqrt{2 x-3}, g(x)=1 / 2 x^{2}+3 / 2 x \geq 0$$

Bobby B.
University of North Texas

Problem 60

Verify that $f$ and $g$ are inverse functions using the composition property.
$$f(x)=x^{2}+1, x \geq 0, g(x)=\sqrt{x-1}$$

Bobby B.
University of North Texas

Problem 61

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.
Exercise 29

Bobby B.
University of North Texas

Problem 62

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.
Exercise 32

Bobby B.
University of North Texas

Problem 63

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.
Exercise 35

Bobby B.
University of North Texas

Problem 64

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.
Exercise 37

Bobby B.
University of North Texas

Problem 65

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.
$$f(x)=\frac{3 x-2}{2 x+5}$$

Bobby B.
University of North Texas

Problem 66

(a) Given $f(x)=2 x^{3}+3 x-4,$ show this function is one-to-one
(b) Determine $\left(f^{-1}(x)\right)^{\prime}(18)$.

Bobby B.
University of North Texas

Problem 67

(a) Given $f(x)=3 x^{5}+2 x^{3}+2,$ show this function is one-to-one.
(b) Determine $\left(f^{-1}(x)\right)^{\prime}(7)$.

Bobby B.
University of North Texas

Problem 68

Prove, using (4) that $\frac{d}{d x}\left(x^{1 / 3}\right)=\frac{1}{3 x^{2 / 3}}$.

Bobby B.
University of North Texas

Problem 69

Consider the function defined by
$$y=f(x)=\left\{\begin{array}{cc}x & 0 \leq x < 1 \\ -x & x \geq 1\end{array}\right.$$
sketch the graph of this function and determine if it is one-to-one.

Bobby B.
University of North Texas

Problem 70

Consider the function defined by
$$y=f(x)=\left\{\begin{array}{cc}-2 x & 0 \leq x < 2 \\ 4 x & x \geq 2\end{array}\right.$$
sketch the graph of this function and determine if it is one-to-one.

Bobby B.
University of North Texas

Problem 71

Will a one-to-one function always be a decreasing or increasing function?

Bobby B.
University of North Texas

Problem 72

Are there functions that are their own inverse?

Bobby B.
University of North Texas

Problem 73

(a) Do even continuous functions have an inverse? (b) Odd functions?

Bobby B.
University of North Texas

Problem 74

Consider $f(x)=\frac{4 x}{x^{2}-9},$ (a) Show that this function is always decreasing.
(b) Does this function have an inverse? Explain.

Bobby B.
University of North Texas

Problem 75

Suppose $f$ and $g$ are inverse functions and have second derivatives. Show that
$$g^{\prime \prime}(x)=-\frac{f^{\prime \prime}(g(x))}{\left(f^{\prime}(g(x))\right)^{3}}$$

Bobby B.
University of North Texas

Problem 76

Using the results of the previous exercise, determine how the concavity of $g$ is related to the concavity of $f$ Hint: there are four cases to consider.

Check back soon!

Problem 77

Prove the inverse function is unique. Hint: assume both $g$ and $h$ are inverse functions of $f$ consider $(g f h)(x)$.

Bobby B.
University of North Texas