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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

f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE
Thanks for this helpfull app
Axmed Reply
Practice Key Terms 1

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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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