# 3.4 Composition of functions  (Page 8/9)

 Page 8 / 9

$h\left(x\right)={\left(\frac{8+{x}^{3}}{8-{x}^{3}}\right)}^{4}$

$h\left(x\right)=\sqrt{2x+6}$

sample: $\begin{array}{l}\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}g\left(x\right)=2x+6\end{array}$

$h\left(x\right)={\left(5x-1\right)}^{3}$

$h\left(x\right)=\sqrt[3]{x-1}$

sample: $\begin{array}{l}\text{\hspace{0.17em}}f\left(x\right)=\sqrt[3]{x}\\ \text{\hspace{0.17em}}g\left(x\right)=\left(x-1\right)\end{array}$

$h\left(x\right)=|{x}^{2}+7|$

$h\left(x\right)=\frac{1}{{\left(x-2\right)}^{3}}$

sample: $\begin{array}{l}\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\\ \text{\hspace{0.17em}}g\left(x\right)=\frac{1}{x-2}\end{array}$

$h\left(x\right)={\left(\frac{1}{2x-3}\right)}^{2}$

$h\left(x\right)=\sqrt{\frac{2x-1}{3x+4}}$

sample: $\begin{array}{l}\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\\ \text{\hspace{0.17em}}g\left(x\right)=\frac{2x-1}{3x+4}\end{array}$

## Graphical

For the following exercises, use the graphs of $\text{\hspace{0.17em}}f,$ shown in [link] , and $\text{\hspace{0.17em}}g,$ shown in [link] , to evaluate the expressions.

$f\left(g\left(3\right)\right)$

$f\left(g\left(1\right)\right)$

2

$g\left(f\left(1\right)\right)$

$g\left(f\left(0\right)\right)$

5

$f\left(f\left(5\right)\right)$

$f\left(f\left(4\right)\right)$

4

$g\left(g\left(2\right)\right)$

$g\left(g\left(0\right)\right)$

0

For the following exercises, use graphs of $\text{\hspace{0.17em}}f\left(x\right),$ shown in [link] , $\text{\hspace{0.17em}}g\left(x\right),$ shown in [link] , and $\text{\hspace{0.17em}}h\left(x\right),$ shown in [link] , to evaluate the expressions.

$g\left(f\left(1\right)\right)$

$g\left(f\left(2\right)\right)$

2

$f\left(g\left(4\right)\right)$

$f\left(g\left(1\right)\right)$

1

$f\left(h\left(2\right)\right)$

$h\left(f\left(2\right)\right)$

4

$f\left(g\left(h\left(4\right)\right)\right)$

$f\left(g\left(f\left(-2\right)\right)\right)$

4

## Numeric

For the following exercises, use the function values for shown in [link] to evaluate each expression.

$x$ $f\left(x\right)$ $g\left(x\right)$
0 7 9
1 6 5
2 5 6
3 8 2
4 4 1
5 0 8
6 2 7
7 1 3
8 9 4
9 3 0

$f\left(g\left(8\right)\right)$

$f\left(g\left(5\right)\right)$

9

$g\left(f\left(5\right)\right)$

$g\left(f\left(3\right)\right)$

4

$f\left(f\left(4\right)\right)$

$f\left(f\left(1\right)\right)$

2

$g\left(g\left(2\right)\right)$

$g\left(g\left(6\right)\right)$

3

For the following exercises, use the function values for shown in [link] to evaluate the expressions.

 $x$ $f\left(x\right)$ $g\left(x\right)$ $-3$ 11 $-8$ $-2$ 9 $-3$ $-1$ 7 0 0 5 1 1 3 0 2 1 $-3$ 3 $-1$ $-8$

$\left(f\circ g\right)\left(1\right)$

$\left(f\circ g\right)\left(2\right)$

11

$\left(g\circ f\right)\left(2\right)$

$\left(g\circ f\right)\left(3\right)$

0

$\left(g\circ g\right)\left(1\right)$

$\left(f\circ f\right)\left(3\right)$

7

For the following exercises, use each pair of functions to find $\text{\hspace{0.17em}}f\left(g\left(0\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(0\right)\right).$

$f\left(x\right)=4x+8,\text{\hspace{0.17em}}g\left(x\right)=7-{x}^{2}$

$f\left(x\right)=5x+7,\text{\hspace{0.17em}}g\left(x\right)=4-2{x}^{2}$

$f\left(g\left(0\right)\right)=27,\text{\hspace{0.17em}}g\left(f\left(0\right)\right)=-94$

$f\left(x\right)=\sqrt{x+4},\text{\hspace{0.17em}}g\left(x\right)=12-{x}^{3}$

$f\left(x\right)=\frac{1}{x+2},\text{\hspace{0.17em}}g\left(x\right)=4x+3$

$f\left(g\left(0\right)\right)=\frac{1}{5},\text{\hspace{0.17em}}g\left(f\left(0\right)\right)=5$

For the following exercises, use the functions $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=3x+5\text{\hspace{0.17em}}$ to evaluate or find the composite function as indicated.

$f\left(g\left(2\right)\right)$

$f\left(g\left(x\right)\right)$

$18{x}^{2}+60x+51$

$g\left(f\left(-3\right)\right)$

$\left(g\circ g\right)\left(x\right)$

$g\circ g\left(x\right)=9x+20$

## Extensions

For the following exercises, use $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt[3]{x-1}.$

Find $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right).\text{\hspace{0.17em}}$ Compare the two answers.

Find $\text{\hspace{0.17em}}\left(f\circ g\right)\left(2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(g\circ f\right)\left(2\right).$

2

What is the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)?$

What is the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)?$

$\left(-\infty ,\infty \right)$

Let $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}.$

1. Find $\text{\hspace{0.17em}}\left(f\circ f\right)\left(x\right).$
2. Is $\text{\hspace{0.17em}}\left(f\circ f\right)\left(x\right)\text{\hspace{0.17em}}$ for any function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ the same result as the answer to part (a) for any function? Explain.

For the following exercises, let $\text{\hspace{0.17em}}F\left(x\right)={\left(x+1\right)}^{5},\text{\hspace{0.17em}}$ $f\left(x\right)={x}^{5},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=x+1.$

True or False: $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=F\left(x\right).$

False

True or False: $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)=F\left(x\right).$

For the following exercises, find the composition when $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}+2\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\ge 0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-2}.$

$\left(f\circ g\right)\left(6\right);\text{\hspace{0.17em}}\left(g\circ f\right)\left(6\right)$

$\left(f\circ g\right)\left(6\right)=6$ ; $\text{\hspace{0.17em}}\left(g\circ f\right)\left(6\right)=6$

$\left(g\circ f\right)\left(a\right);\text{\hspace{0.17em}}\left(f\circ g\right)\left(a\right)$

$\left(f\circ g\right)\left(11\right);\text{\hspace{0.17em}}\left(g\circ f\right)\left(11\right)$

$\left(f\circ g\right)\left(11\right)=11\text{\hspace{0.17em}},\text{\hspace{0.17em}}\left(g\circ f\right)\left(11\right)=11$

## Real-world applications

The function $\text{\hspace{0.17em}}D\left(p\right)\text{\hspace{0.17em}}$ gives the number of items that will be demanded when the price is $\text{\hspace{0.17em}}p.\text{\hspace{0.17em}}$ The production cost $\text{\hspace{0.17em}}C\left(x\right)\text{\hspace{0.17em}}$ is the cost of producing $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items. To determine the cost of production when the price is \$6, you would do which of the following?

1. Evaluate $\text{\hspace{0.17em}}D\left(C\left(6\right)\right).$
2. Evaluate $\text{\hspace{0.17em}}C\left(D\left(6\right)\right).$
3. Solve $\text{\hspace{0.17em}}D\left(C\left(x\right)\right)=6.$
4. Solve $\text{\hspace{0.17em}}C\left(D\left(p\right)\right)=6.$

The function $\text{\hspace{0.17em}}A\left(d\right)\text{\hspace{0.17em}}$ gives the pain level on a scale of 0 to 10 experienced by a patient with $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes is modeled by $\text{\hspace{0.17em}}m\left(t\right).\text{\hspace{0.17em}}$ Which of the following would you do in order to determine when the patient will be at a pain level of 4?

1. Evaluate $\text{\hspace{0.17em}}A\left(m\left(4\right)\right).$
2. Evaluate $\text{\hspace{0.17em}}m\left(A\left(4\right)\right).$
3. Solve $\text{\hspace{0.17em}}A\left(m\left(t\right)\right)=4.$
4. Solve $\text{\hspace{0.17em}}m\left(A\left(d\right)\right)=4.$

c

A store offers customers a 30% discount on the price $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function $\text{\hspace{0.17em}}P\left(x\right)\text{\hspace{0.17em}}$ that computes the final price of the item in terms of the original price $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ (Hint: Use function composition to find your answer.)

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to $\text{\hspace{0.17em}}r\left(t\right)=25\sqrt{t+2},\text{\hspace{0.17em}}$ find the area of the ripple as a function of time. Find the area of the ripple at $\text{\hspace{0.17em}}t=2.$

$A\left(t\right)=\pi {\left(25\sqrt{t+2}\right)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}A\left(2\right)=\pi {\left(25\sqrt{4}\right)}^{2}=2500\pi$ square inches

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula $\text{\hspace{0.17em}}r\left(t\right)=2t+1,\text{\hspace{0.17em}}$ express the area burned as a function of time, $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ (minutes).

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

$A\left(5\right)=\pi {\left(2\left(5\right)+1\right)}^{2}=121\pi \text{\hspace{0.17em}}$ square units

The radius $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ in inches, of a spherical balloon is related to the volume, $\text{\hspace{0.17em}}V,\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}r\left(V\right)=\sqrt[3]{\frac{3V}{4\pi }}.\text{\hspace{0.17em}}$ Air is pumped into the balloon, so the volume after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given by $\text{\hspace{0.17em}}V\left(t\right)=10+20t.$

1. Find the composite function $\text{\hspace{0.17em}}r\left(V\left(t\right)\right).$
2. Find the exact time when the radius reaches 10 inches.

The number of bacteria in a refrigerated food product is given by $N\left(T\right)=23{T}^{2}-56T+1,\text{\hspace{0.17em}}$ $3 where $\text{\hspace{0.17em}}T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T\left(t\right)=5t+1.5,$ where $t$ is the time in hours.

1. Find the composite function $\text{\hspace{0.17em}}N\left(T\left(t\right)\right).$
2. Find the time (round to two decimal places) when the bacteria count reaches 6752.

a. $\text{\hspace{0.17em}}N\left(T\left(t\right)\right)=23{\left(5t+1.5\right)}^{2}-56\left(5t+1.5\right)+1;\text{\hspace{0.17em}}$ b. 3.38 hours

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