# 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

Find that number sum and product of all the divisors of 360
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
give me treganamentry question
Solve 2cos x + 3sin x = 0.5
madras university algebra questions papers first year B. SC. maths
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
give me some important question in tregnamentry
Anshuman