# 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 #### Questions & Answers (1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin\$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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