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

h ( x ) = 2 x + 6

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

h ( x ) = ( 5 x 1 ) 3

h ( x ) = x 1 3

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

h ( x ) = | x 2 + 7 |

h ( x ) = 1 ( x 2 ) 3

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

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

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

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

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.

f ( g ( 3 ) )

f ( g ( 1 ) )

2

g ( f ( 1 ) )

g ( f ( 0 ) )

5

f ( f ( 5 ) )

f ( f ( 4 ) )

4

g ( g ( 2 ) )

g ( g ( 0 ) )

0

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.

g ( f ( 1 ) )

g ( f ( 2 ) )

2

f ( g ( 4 ) )

f ( g ( 1 ) )

1

f ( h ( 2 ) )

h ( f ( 2 ) )

4

f ( g ( h ( 4 ) ) )

f ( g ( f ( 2 ) ) )

4

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

f ( g ( 8 ) )

f ( g ( 5 ) )

9

g ( f ( 5 ) )

g ( f ( 3 ) )

4

f ( f ( 4 ) )

f ( f ( 1 ) )

2

g ( g ( 2 ) )

g ( g ( 6 ) )

3

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

( f g ) ( 1 )

( f g ) ( 2 )

11

( g f ) ( 2 )

( g f ) ( 3 )

0

( g g ) ( 1 )

( f f ) ( 3 )

7

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

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

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

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

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

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

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 ( 2 ) )

f ( g ( x ) )

18 x 2 + 60 x + 51

g ( f ( 3 ) )

( g g ) ( x )

g g ( x ) = 9 x + 20

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.

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

2

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

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

( , )

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.

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

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

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

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

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

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

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.

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

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.)

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

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).

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

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.

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

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
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