Section 1
Operations on Functions
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=3 x+4} \\ {g(x)=5+x}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=x^{2}+3} \\ {g(x)=x-4}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(-1,9),(4,7)\}} \\ {g=\{(-5,4),(7,12),(4,-1)\}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(0,-7),(1,2),(2,-1)\}} \\ {g=\{(-1,10),(2,0)\}}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=2 x} \\ {h(x)=3 x-4}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=x+5} \\ {h(x)=x^{2}+6}\end{array}$$
If $f(x)=3 x, g(x)=x+7,$ and $h(x)=x^{2},$ find each value.$$f[g(3)]$$
If $f(x)=3 x, g(x)=x+7,$ and $h(x)=x^{2},$ find each value.$$g[h(-2)]$$
If $f(x)=3 x, g(x)=x+7,$ and $h(x)=x^{2},$ find each value.$$h[h(1)]$$
For Exercises $10-13,$ use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs $\$ 49.99,$ but is on sale at a 25$\%$ discount. She also has a $\$ 5$ coupon she can use.Express the price of the CD after the discount and the price of the CD after the coupon. Let $x$ represent the price of the CD, $p(x)$ represent the price after the 25$\%$ discount, and $c(x)$ represent the price after the coupon.
For Exercises $10-13,$ use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs $\$ 49.99,$ but is on sale at a 25$\%$ discount. She also has a $\$ 5$ coupon she can use.Find $c[p(x)]$ and explain what this value represents.
For Exercises $10-13,$ use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs $\$ 49.99,$ but is on sale at a 25$\%$ discount. She also has a $\$ 5$ coupon she can use.Find $p[c(x)]$ and explain what this value represents.
For Exercises $10-13,$ use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs $\$ 49.99,$ but is on sale at a 25$\%$ discount. She also has a $\$ 5$ coupon she can use.Which method results in the lower sale price? Explain your reasoning.
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=x+9} \\ {g(x)=x-9}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=2 x-3} \\ {g(x)=4 x+9}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=2 x^{2}} \\ {g(x)=8-x}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=x^{2}+6 x+9} \\ {g(x)=2 x+6}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=x^{2}-1} \\ {g(x)=\frac{x}{x+1}}\end{array}$$
Find $(f+g)(x),(f-g)(x),(f \cdot g)(x),$ and $\left(\frac{f}{g}\right)$ for each $f(x)$ and $g(x)$$$\begin{array}{l}{f(x)=x^{2}-x-6} \\ {g(x)=\frac{x-3}{x+2}}\end{array}$$
For Exercises 20 and $21,$ use the following information. Carlos is walking on a moving walkway. His speed is given by the function $C(x)=3 x^{2}+3 x-4,$ and the speed of the walkway is $W(x)=x^{2}-4 x+7$What is his total speed as he walks along the moving walkway?
For Exercises 20 and $21,$ use the following information. Carlos is walking on a moving walkway. His speed is given by the function $C(x)=3 x^{2}+3 x-4,$ and the speed of the walkway is $W(x)=x^{2}-4 x+7$Carlos turned around because he left his cell phone at a restaurant. What was his speed as he walked against the moving walkway?
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(1,1),(0,-3)\}} \\ {g=\{(1,0),(-3,1),(2,1)\}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(1,2),(3,4),(5,4)\}} \\ {g=\{(2,5),(4,3)\}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(3,8),(4,0),(6,3),(7,-1)\}} \\ {g=\{(0,4),(8,6),(3,6),(-1,8)\}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(4,5),(6,5),(8,12),(10,12)\}} \\ {g=\{4,6),(2,4),(6,8),(8,10) \}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(2,5),(3,9),(-4,1)\}} \\ {g=\{(5,-4),(8,3),(2,-2)\}}\end{array}$$
For each pair of functions, find $f \circ g$ and $g \circ f,$ if they exist.$$\begin{array}{l}{f=\{(7,0),(-5,3),(8,3),(-9,2)\}} \\ {g=\{(2,-5),(1,0),(2,-9),(3,6)\}}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=4 x} \\ {h(x)=2 x-1}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=-5 x} \\ {h(x)=-3 x+1}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=x+2} \\ {h(x)=x^{2}}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=x-4} \\ {h(x)=3 x^{2}}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=2 x} \\ {h(x)=x^{3}+x^{2}+x+1}\end{array}$$
Find $[g \circ h](x)$ and $[h \circ g](x)$$$\begin{array}{l}{g(x)=x+1} \\ {h(x)=2 x^{2}-5 x+8}\end{array}$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$f[g(-1)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$h[g(4)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$g[f(5)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$f[h(-4)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$g[g(7)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$f[f(-3)]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$h\left[f\left(\frac{1}{4}\right)\right]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$g\left[h\left(-\frac{1}{2}\right)\right]$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$[g \circ(f \circ h)](3)$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$[f \circ(h \circ g)](3)$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$[h \circ(g \circ f)](2)$$
If $f(x)=4 x, g(x)=2 x-1,$ and $h(x)=x^{2}+1,$ find each value.$$[f \circ(g \circ h)](2)$$
For Exercises 46 and $47,$ use the following information. From 1990 to $2002,$ the number of births $b(x)$ in the United States can be modeled by the function $b(x)=-8 x+4045,$ and the number of deaths $d(x)$ can be modeled by the function $d(x)=24 x+2160,$ where $x$ is the number of years since 1990 and $b(x)$ and $d(x)$ are in thousands.The net increase in population $P$ is the number of births per year minus the number of deaths per year, or $P=b-d$ . Write an expression that can be used to model the population increase in the U.S. from 1990 to 2002 in function notation.
For Exercises 46 and $47,$ use the following information. From 1990 to $2002,$ the number of births $b(x)$ in the United States can be modeled by the function $b(x)=-8 x+4045,$ and the number of deaths $d(x)$ can be modeled by the function $d(x)=24 x+2160,$ where $x$ is the number of years since 1990 and $b(x)$ and $d(x)$ are in thousands.Assume that births and deaths continue at the same rates. Estimate the net increase in population in $2015 .$
For Exercises $48-50,$ use the following information. Liluye wants to buy a pair of inline skates that are on sale for 30$\%$ off the original price of $\$ 149 .$ The sales tax is 5.75$\% .$Express the price of the inline skates after the discount and the price of the inline skates after the sales tax using function notation. Let $x$ represent the price of the inline skates $p(x)$ represent the price after the 30$\%$ discount, and $s(x)$ represent the price after the sales tax.
For Exercises $48-50,$ use the following information. Liluye wants to buy a pair of inline skates that are on sale for 30$\%$ off the original price of $\$ 149 .$ The sales tax is 5.75$\% .$Which composition of functions represents the price of the inline skates, $p[s(x)]$ or $s[p(x)]$ ? Explain your reasoning.
For Exercises $48-50,$ use the following information. Liluye wants to buy a pair of inline skates that are on sale for 30$\%$ off the original price of $\$ 149 .$ The sales tax is 5.75$\% .$How much will Liluye pay for the inline skates?
FINANCE Regina pays $\$ 50$ each month on a credit card that charges 1.6$\%$ interest monthly. She has a balance of $\$ 700 .$ The balance at the beginning of the $n$ th month is given by $f(n)=f(n-1)+0.016 f(n-1)-50 .$ Find the balance at the beginning of the first five months. No additional charges are made on the card. (Hint: $f(1)=700 )$
OPEN ENDED Write a set of ordered pairs for functions $f$ and $g,$ given that $f \circ g=\{(4,3),(-1,9),(-2,7)\}$
FIND THE ERROR Danette and Marquan are trying to find $[g \circ f](3)$ for $f(x)=x^{2}+4 x+5$ and $g(x)=x-7 .$ Who is correct? Explain your reasoning.
If $f(0)=4$ and $f(x+1)=3 f(x)-2,$ find $f(4)$
Refer to the information on page 384 to explain how combining functions can be important to business. Describe how to write a new function that represents the profit, using the revenue and cost functions. What are the benefits of combining two functions into one function?
ACT/SAT What is the value of $f(g(6))$ if $f(x)=2 x+4$ and $g(x)=x^{2}+5 ?$A 38B 43C 86D 261
REVIEW If $g(x)=x^{2}+9 x+21$ and $h(x)=2(x+5)^{2},$ which is an equivalent form of $h(x)-g(x) ?$$$\begin{array}{l}{\mathbf{F}-x^{2}-11 x-29} \\ {\mathbf{G} x^{2}+11 x+29} \\ {\mathbf{H} x+4} \\ {\mathbf{J} \quad x^{2}+7 x+11}\end{array}$$
List all of the possible rational zeros of each function.$$r(x)=x^{2}-6 x+8$$
List all of the possible rational zeros of each function.$$f(x)=4 x^{3}-2 x^{2}+6$$
List all of the possible rational zeros of each function.$$g(x)=9 x^{2}-1$$
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function.$$f(x)=7 x^{4}+3 x^{3}-2 x^{2}-x+1$$
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function.$$g(x)=2 x^{4}-x^{3}-3 x+7$$
CHEMISTRY The mass of a proton is about $1.67 \times 10^{-27}$ kilogram. The mass of an electron is about $9.11 \times 10^{-31}$ kilogram. About how many times as massive as an electron is a proton?
Solve each equation or formula for the specified variable.$$2 x-3 y=6, \text { for } x$$
Solve each equation or formula for the specified variable.$$4 x^{2}-5 x y+2=3, \text { for } y$$
Solve each equation or formula for the specified variable.$$3 x+7 x y=-2, \text { for } x$$
Solve each equation or formula for the specified variable.$$I=p r t, \text { for } t$$
Solve each equation or formula for the specified variable.$$C=\frac{5}{9}(F-32), \text { for } F$$
Solve each equation or formula for the specified variable.$$F=G \frac{M m}{r^{2}}, \text { for } m$$