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Intermediate Algebra

Julie Miller, Molly O'Neill, Nancy Hyde

Chapter 8

Exponential and Logarithmic Functions and Applications - all with Video Answers

Educators

AG

Section 1

Algebra of Functions and Composition

00:54

Problem 1

a. Given the functions $f$ and $g,$ the function $(f+g)(x)=$___ +____.
b. Given the functions $f$ and $g$, the function $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{\square},$ provided___$\neq 0$.
c. The function $f$ composed with $g$ is defined by the rule $(f \circ g)(x)=$___.

AG
Ankit Gupta
Numerade Educator
01:49

Problem 2

Given $f(x)=x^{2}$ and $g(x)=2 x-3,$ find
a. $f(-2)$
b. $g(-2)$
c. $(f+g)(-2)$

AG
Ankit Gupta
Numerade Educator
00:51

Problem 3

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(f+g)(x)$$

AG
Ankit Gupta
Numerade Educator
00:38

Problem 4

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(f-g)(x)$$

AG
Ankit Gupta
Numerade Educator
00:37

Problem 5

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(g-f)(x)$$

AG
Ankit Gupta
Numerade Educator
00:34

Problem 6

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(f+h)(x)$$

AG
Ankit Gupta
Numerade Educator
00:41

Problem 7

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(f \cdot h)(x)$$

AG
Ankit Gupta
Numerade Educator
00:37

Problem 8

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(h \cdot k)(x)$$

AG
Ankit Gupta
Numerade Educator
00:51

Problem 9

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(g \cdot f)(x)$$

AG
Ankit Gupta
Numerade Educator
00:35

Problem 10

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$(f \cdot k)(x)$$

AG
Ankit Gupta
Numerade Educator
00:43

Problem 11

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$\left(\frac{h}{f}\right)(x)$$

AG
Ankit Gupta
Numerade Educator
00:30

Problem 12

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$\left(\frac{g}{f}\right)(x)$$

AG
Ankit Gupta
Numerade Educator
00:27

Problem 13

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$\left(\frac{f}{g}\right)(x)$$

AG
Ankit Gupta
Numerade Educator
00:28

Problem 14

Refer to the functions defined below.
$$f(x)=x+4 \quad g(x)=2 x^{2}+4 x \quad h(x)=x^{2}+1 \quad k(x)=\frac{1}{x}$$
Find the indicated functions.
$$\left(\frac{f}{h}\right)(x)$$

AG
Ankit Gupta
Numerade Educator
00:54

Problem 15

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(f \circ g)(x)$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 16

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(f \circ k)(x)$$

AG
Ankit Gupta
Numerade Educator
01:18

Problem 17

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(g \circ f)(x)$$

AG
Ankit Gupta
Numerade Educator
00:53

Problem 18

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(k \cdot f)(x)$$

AG
Ankit Gupta
Numerade Educator
00:45

Problem 19

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(k \circ h)(x)$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 20

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(h \cdot k)(x)$$

AG
Ankit Gupta
Numerade Educator
00:35

Problem 21

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(k \circ g)(x)$$

AG
Ankit Gupta
Numerade Educator
01:11

Problem 22

Find the indicated functions. Use $f, g, h,$ and $k$ .
$$(g \circ k)(x)$$

AG
Ankit Gupta
Numerade Educator
01:39

Problem 23

Based on your answers to Exercises 15 and $17,$ is it true in general that $(f \circ g)(x)=(g \circ f)(x) ?$

AG
Ankit Gupta
Numerade Educator
01:25

Problem 24

Based on your answers to Exercises 16 and $18,$ is it true in general that $(f \circ k)(x)=(k \cdot f)(x) ?$

AG
Ankit Gupta
Numerade Educator
01:39

Problem 25

Find $(f \circ g)(x)$ and $(g \circ f)(x)$.
$$f(x)=x^{2}-3 x+1, g(x)=5 x$$

AG
Ankit Gupta
Numerade Educator
01:48

Problem 26

Find $(f \circ g)(x)$ and $(g \circ f)(x)$.
$$f(x)=3 x^{2}+8, g(x)=2 x-4$$

AG
Ankit Gupta
Numerade Educator
01:31

Problem 27

Find $(f \circ g)(x)$ and $(g \circ f)(x)$.
$$f(x)=|x|, g(x)=x^{3}-1$$

AG
Ankit Gupta
Numerade Educator
01:22

Problem 28

Find $(f \circ g)(x)$ and $(g \circ f)(x)$.
$$f(x)=\frac{1}{x+2}, g(x)=|x+2|$$

AG
Ankit Gupta
Numerade Educator

Problem 29

For $h(x)=5 x-4$
find $(h \circ h)(x)$

Check back soon!
01:22

Problem 30

For $k(x)=-x^{2}+1$
find $(k=k)(x)$

AG
Ankit Gupta
Numerade Educator
00:54

Problem 31

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(m+r)(0)$$

AG
Ankit Gupta
Numerade Educator
00:51

Problem 32

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(n \cdot p)(0)$$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 33

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(m+r)(-4)$$

AG
Ankit Gupta
Numerade Educator
00:55

Problem 34

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(n-m)(4)$$

AG
Ankit Gupta
Numerade Educator
00:55

Problem 35

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(r \circ n)(3)$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 36

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(n \circ r)(5)$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 37

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(p \circ m)(-1)$$

AG
Ankit Gupta
Numerade Educator
00:45

Problem 38

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(m \circ n)(5)$$

AG
Ankit Gupta
Numerade Educator
01:05

Problem 39

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(m \circ p)(2)$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 40

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(r \circ m)(2)$$

AG
Ankit Gupta
Numerade Educator
01:14

Problem 41

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(r+p)(-3)$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 42

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(n+p)(-2)$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 43

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(m \circ p)(-2)$$

AG
Ankit Gupta
Numerade Educator
01:19

Problem 44

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$(r \circ m)(-2)$$

AG
Ankit Gupta
Numerade Educator
00:51

Problem 45

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$\left(\frac{r}{n}\right)(12)$$

AG
Ankit Gupta
Numerade Educator
00:41

Problem 46

Refer to the functions defined below.
$$m(x)=x^{3} \quad n(x)=x-3 \quad r(x)=\sqrt{x+4} \quad p(x)=\frac{1}{x+2}$$
Find each function value if possible.
$$\left(\frac{n}{m}\right)(2)$$

AG
Ankit Gupta
Numerade Educator
00:27

Problem 47

Approximate each function value from the graph, if possible.
$$f(-4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:21

Problem 48

Approximate each function value from the graph, if possible.
$$f(1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:28

Problem 49

Approximate each function value from the graph, if possible.
$$g(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:26

Problem 50

Approximate each function value from the graph, if possible.
$$g(3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:47

Problem 51

Approximate each function value from the graph, if possible.
$$(f+g)(2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:42

Problem 52

Approximate each function value from the graph, if possible.
$$(g-f)(3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:41

Problem 53

Approximate each function value from the graph, if possible.
$$(f \cdot g)(-1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:41

Problem 54

Approximate each function value from the graph, if possible.
$$(g \cdot f)(-4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:35

Problem 55

Approximate each function value from the graph, if possible.
$$\left(\frac{g}{f}\right)(0)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:37

Problem 56

Approximate each function value from the graph, if possible.
$$\left(\frac{f}{g}\right)(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:36

Problem 57

Approximate each function value from the graph, if possible.
$$\left(\frac{f}{g}\right)(0)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:37

Problem 58

Approximate each function value from the graph, if possible.
$$\left(\frac{g}{f}\right)(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:41

Problem 59

Approximate each function value from the graph, if possible.
$$(g \circ f)(-1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:34

Problem 60

Approximate each function value from the graph, if possible.
$$(f \circ g)(0)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:38

Problem 61

Approximate each function value from the graph, if possible.
$$(f \circ g)(-4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:39

Problem 62

Approximate each function value from the graph, if possible.
$$(g \circ f)(-4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:35

Problem 63

Approximate each function value from the graph, if possible.
$$(g \circ g)(2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:32

Problem 64

Approximate each function value from the graph, if possible.
$$(f \circ f)(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:31

Problem 65

Approximate each function value from the graph, if possible.
$$a(-3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:33

Problem 66

Approximate each function value from the graph, if possible.
$$a(1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:29

Problem 67

Approximate each function value from the graph, if possible.
$$b(-1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:27

Problem 68

Approximate each function value from the graph, if possible.
$$b(3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:46

Problem 69

Approximate each function value from the graph, if possible.
$$(a-b)(-1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:40

Problem 70

Approximate each function value from the graph, if possible.
$$(a+b)(0)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:46

Problem 71

Approximate each function value from the graph, if possible.
$$(b \cdot a)(1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:51

Problem 72

Approximate each function value from the graph, if possible.
$$(a \cdot b)(2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:40

Problem 73

Approximate each function value from the graph, if possible.
$$(b \bullet a)(0)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
01:00

Problem 74

Approximate each function value from the graph, if possible.
$$(a \cdot b)(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:46

Problem 75

Approximate each function value from the graph, if possible.
$$(a \circ b)(-4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:38

Problem 76

Approximate each function value from the graph, if possible.
$$(b \cdot a)(-3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:49

Problem 77

Approximate each function value from the graph, if possible.
$$\left(\frac{b}{a}\right)(3)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:42

Problem 78

Approximate each function value from the graph, if possible.
$$\left(\frac{a}{b}\right)(4)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:36

Problem 79

Approximate each function value from the graph, if possible.
$$(a \circ a)(-2)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
00:37

Problem 80

Approximate each function value from the graph, if possible.
$$(b \cdot b)(1)$$
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
01:54

Problem 81

The cost in dollars of producing $x$ toy cars is $C(x)=2.2 x+1 .$ The revenue for selling $x$ toy cars is $R(x)=5.98 x .$ To calculate profit, subtract the cost from the revenue.
a. Write and simplify a function $P$ that represents profit in terms of $x$.
b. Find the profit of producing 50 toy ears.

AG
Ankit Gupta
Numerade Educator
02:16

Problem 82

The cost in dollars of producing $x$ lawn chairs is $C(x)=2.5 x+10.1 .$ The revenue for selling $x$ lawn chairs is $R(x)=6.99 x .$ To calculate profit, subtract the cost from the revenue.
a. Write and simplify a function $P$ that represents profit in terms of $x$.
b. Find the profit in producing 100 lawn chairs.

AG
Ankit Gupta
Numerade Educator
03:21

Problem 83

The functions defined by $D(t)=0.925 t+26.958$ and $R(t)=0.725 t+20.558$ approximate the amount of child support (in billions of dollars) that was due $D(t)$ and the amount of child support actually received $R(t)$ in the United States for a selected number of years. In each case, $t=0$ represents the first year of the study.
a. Find the function defined by $F(t)=D(t)-R(t) .$ What does $F(t)$ represent in the context of this problem?
b. Find $F(4)$. What does this function value represent in the context of this problem?

AG
Ankit Gupta
Numerade Educator
04:10

Problem 84

The rural and urban populations in the South (in the United States) between the years 1900 and 1970 can be modeled by the following:
$$\begin{array}{l}r(t)=-3.497 t^{2}+266.2 t+20.220 \\u(t)=0.0566 r^{3}+0.952 t^{2}+177.8 t+4593\end{array}$$
The variable $t$ represents the number of years since $1900, r(t)$ represents the rural population in thousands, and $u(t)$ represents the urban population in thousands,
a. Find the function defined by $T(t)=r(t)+u(t)$ What does $T(t)$ represent in the context of this problem?
b. Use the function $T$ to approximate the total population
(GRAPH CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
02:21

Problem 85

Joe rides a bicycle and his wheels revolve at
80 revolutions per minute (rpm). Therefore, the total number of revolutions, $r$, is given by $r(t)=80 t,$ where $t$ is the time in minutes. For each revolution of the wheels of the bike, he travels approximately 7 ft. Therefore, the total distance he travels $D(r)$ (in feet) depends on the total number of revolutions $r$ according to the function $D(r)=7 r$
a. Find $(D \circ r)(t)$ and interpret its meaning in the context of this problem.
b. Find Joe's total distance in feet after 10 min.

AG
Ankit Gupta
Numerade Educator
02:35

Problem 86

The area of a square is given by the function $a(x)=x^{2},$ where $x$ is the length of the sides of the square, If carpeting costs $\$ 9.95$ per square yard, then the cost $C(a)$ (in dollars) to carpet a square room is given by $C(a)=9.95 a,$ where $a$ is the area of the room in square yards.
a. Find $(C \circ a)(x)$ and interpret its meaning in the context of this problem.
b. Find the cost to carpet a square room if its floor dimensions are 15 yd by 15 yd.

AG
Ankit Gupta
Numerade Educator