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h ( x ) = ( 8 + x 3 8 x 3 ) 4

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h ( x ) = 2 x + 6

sample: f ( x ) = x g ( x ) = 2 x + 6

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h ( x ) = ( 5 x 1 ) 3

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h ( x ) = x 1 3

sample: f ( x ) = x 3 g ( x ) = ( x 1 )

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h ( x ) = 1 ( x 2 ) 3

sample: f ( x ) = x 3 g ( x ) = 1 x 2

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h ( x ) = ( 1 2 x 3 ) 2

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h ( x ) = 2 x 1 3 x + 4

sample: f ( x ) = x g ( x ) = 2 x 1 3 x + 4

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Graphical

For the following exercises, use the graphs of f , shown in [link] , and g , shown in [link] , to evaluate the expressions.

Graph of a function.
Graph of a function.

For the following exercises, use graphs of f ( x ) , shown in [link] , g ( x ) , shown in [link] , and h ( x ) , shown in [link] , to evaluate the expressions.

Graph of a parabola.
Graph of a square root function.

f ( g ( f ( 2 ) ) )

4

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Numeric

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

x f ( x ) g ( x )
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

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

x f ( x ) g ( x )
−3 11 −8
−2 9 −3
−1 7 0
0 5 1
1 3 0
2 1 −3
3 −1 −8

For the following exercises, use each pair of functions to find f ( g ( 0 ) ) and g ( f ( 0 ) ) .

f ( x ) = 4 x + 8 , g ( x ) = 7 x 2

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f ( x ) = 5 x + 7 , g ( x ) = 4 2 x 2

f ( g ( 0 ) ) = 27 , g ( f ( 0 ) ) = 94

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f ( x ) = x + 4 , g ( x ) = 12 x 3

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f ( x ) = 1 x + 2 , g ( x ) = 4 x + 3

f ( g ( 0 ) ) = 1 5 , g ( f ( 0 ) ) = 5

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For the following exercises, use the functions f ( x ) = 2 x 2 + 1 and g ( x ) = 3 x + 5 to evaluate or find the composite function as indicated.

f ( g ( x ) )

18 x 2 + 60 x + 51

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( g g ) ( x )

g g ( x ) = 9 x + 20

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Extensions

For the following exercises, use f ( x ) = x 3 + 1 and g ( x ) = x 1 3 .

Find ( f g ) ( x ) and ( g f ) ( x ) . Compare the two answers.

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Find ( f g ) ( 2 ) and ( g f ) ( 2 ) .

2

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What is the domain of ( g f ) ( x ) ?

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What is the domain of ( f g ) ( x ) ?

( , )

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Let f ( x ) = 1 x .

  1. Find ( f f ) ( x ) .
  2. Is ( f f ) ( x ) for any function f the same result as the answer to part (a) for any function? Explain.
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For the following exercises, let F ( x ) = ( x + 1 ) 5 , f ( x ) = x 5 , and g ( x ) = x + 1.

True or False: ( g f ) ( x ) = F ( x ) .

False

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True or False: ( f g ) ( x ) = F ( x ) .

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For the following exercises, find the composition when f ( x ) = x 2 + 2 for all x 0 and g ( x ) = x 2 .

( f g ) ( 6 ) ; ( g f ) ( 6 )

( f g ) ( 6 ) = 6 ; ( g f ) ( 6 ) = 6

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( g f ) ( a ) ; ( f g ) ( a )

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( f g ) ( 11 ) ; ( g f ) ( 11 )

( f g ) ( 11 ) = 11 , ( g f ) ( 11 ) = 11

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Real-world applications

The function D ( p ) gives the number of items that will be demanded when the price is p . The production cost C ( x ) is the cost of producing x items. To determine the cost of production when the price is $6, you would do which of the following?

  1. Evaluate D ( C ( 6 ) ) .
  2. Evaluate C ( D ( 6 ) ) .
  3. Solve D ( C ( x ) ) = 6.
  4. Solve C ( D ( p ) ) = 6.
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The function A ( d ) gives the pain level on a scale of 0 to 10 experienced by a patient with d milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after t minutes is modeled by m ( t ) . Which of the following would you do in order to determine when the patient will be at a pain level of 4?

  1. Evaluate A ( m ( 4 ) ) .
  2. Evaluate m ( A ( 4 ) ) .
  3. Solve A ( m ( t ) ) = 4.
  4. Solve m ( A ( d ) ) = 4.

c

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

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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 r ( t ) = 25 t + 2 , find the area of the ripple as a function of time. Find the area of the ripple at t = 2.

A ( t ) = π ( 25 t + 2 ) 2 and A ( 2 ) = π ( 25 4 ) 2 = 2500 π square inches

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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 r ( t ) = 2 t + 1 , express the area burned as a function of time, t (minutes).

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

A ( 5 ) = π ( 2 ( 5 ) + 1 ) 2 = 121 π square units

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The radius r , in inches, of a spherical balloon is related to the volume, V , by r ( V ) = 3 V 4 π 3 . Air is pumped into the balloon, so the volume after t seconds is given by V ( t ) = 10 + 20 t .

  1. Find the composite function r ( V ( t ) ) .
  2. Find the exact time when the radius reaches 10 inches.
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The number of bacteria in a refrigerated food product is given by N ( T ) = 23 T 2 56 T + 1 , 3 < T < 33 , where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T ( t ) = 5 t + 1.5 , where t is the time in hours.

  1. Find the composite function N ( T ( t ) ) .
  2. Find the time (round to two decimal places) when the bacteria count reaches 6752.

a. N ( T ( t ) ) = 23 ( 5 t + 1.5 ) 2 56 ( 5 t + 1.5 ) + 1 ; b. 3.38 hours

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Questions & Answers

hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
Practice Key Terms 1

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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