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This module describes composite functions in Algebra.

You are working in the school cafeteria, making peanut butter sandwiches for today’s lunch.

  • The more classes the school has, the more children there are.
  • The more children there are, the more sandwiches you have to make.
  • The more sandwiches you have to make, the more pounds (lbs) of peanut butter you will use.
  • The more peanut butter you use, the more money you need to budget for peanut butter.

...and so on. Each sentence in this little story is a function. Mathematically, if c size 12{c} {} is the number of classes and h size 12{h} {} is the number of children, then the first sentence asserts the existence of a function h ( c ) size 12{h \( c \) } {} .

The principal walks up to you at the beginning of the year and says “We’re considering expanding the school. If we expand to 70 classes, how much money do we need to budget? What if we expand to 75? How about 80?” For each of these numbers, you have to calculate each number from the previous one, until you find the final budget number.

# Classes # Children # Sandwiches lb. $$

But going through this process each time is tedious. What you want is one function that puts the entire chain together: “You tell me the number of classes, and I will tell you the budget.”

# Classes $$

This is a composite function —a function that represents in one function, the results of an entire chain of dependent functions . Since such chains are very common in real life, finding composite functions is a very important skill.

How do you make a composite function?

We can consider how to build composite functions into the function game that we played on the first day. Suppose Susan takes any number you give her, quadruples it, and adds 6. Al takes any number you give him and divides it by 2. Mathematically, we can represent the two functions like this:

S ( x ) = 4x + 6 size 12{S \( x \) =4x+6} {}
A ( x ) = x 2

To create a chain like the one above, we give a number to Susan; she acts on it, and gives the resulting number to Al; and he then acts on it and hands back a third number.

3 Susan S ( 3 ) = 18 Al A ( 18 ) = 9 size 12{3 rightarrow ital "Susan" rightarrow S \( 3 \) ="18" rightarrow ital "Al" rightarrow A \( "18" \) =9} {}

In this example, we are plugging S ( 3 ) size 12{S \( 3 \) } {} —in other words, 18— into Al’s function. In general, for any x size 12{x} {} that comes in, we are plugging S ( x ) size 12{S \( x \) } {} into A ( x ) size 12{A \( x \) } {} . So we could represent the entire process as A ( S ( x ) ) size 12{A \( S \( x \) \) } {} . This notation for composite functions is really nothing new: it means that you are plugging S ( x ) size 12{S \( x \) } {} into the A function.

But in this case, recall that S ( x ) = 4x + 6 size 12{S \( x \) =4x+6} {} . So we can write:

A ( S ( x ) ) = S ( x ) 2 = 4x + 6 2 = 2x + 3

What happened? We’ve just discovered a shortcut for the entire process. When you perform the operation A ( S ( x ) ) size 12{A \( S \( x \) \) } {} —that is, when you perform the Al function on the result of the Susan function—you are, in effect, doubling and adding 3. For instance, we saw earlier that when we started with a 3, we ended with a 9. Our composite function does this in one step:

3 2x + 3 9 size 12{3 rightarrow 2x+3 rightarrow 9} {}

Understanding the meaning of composite functions requires real thought. It requires understanding the idea that this variable depends on that variable, which in turn depends on the other variable; and how that idea is translated into mathematics. Finding composite functions, on the other hand, is a purely mechanical process—it requires practice, but no creativity. Whenever you are asked for f ( g ( x ) ) size 12{f \( g \( x \) \) } {} , just plug the g ( x ) size 12{g \( x \) } {} function into the f ( x ) size 12{f \( x \) } {} function and then simplify.

Building and testing a composite function

f ( x ) = x 2 4x size 12{f \( x \) =x rSup { size 8{2} } - 4x} {}

g ( x ) = x + 2 size 12{g \( x \) =x+2} {}

What is f ( g ( x ) ) size 12{f \( g \( x \) \) } {} ?

  • To find the composite, plug g ( x ) size 12{g \( x \) } {} into f ( x ) size 12{f \( x \) } {} , just as you would with any number.

f ( g ( x ) ) = ( x + 2 ) 2 4 ( x + 2 ) size 12{f \( g \( x \) \) = \( x+2 \) rSup { size 8{2} } - 4 \( x+2 \) } {}

  • Then simplify.

f ( g ( x ) ) = ( x 2 + 4x + 4 ) ( 4x + 8 ) size 12{f \( g \( x \) \) = \( x"" lSup { size 8{2} } +4x+4 \) - \( 4x+8 \) } {}

f ( g ( x ) ) = x 2 4 size 12{f \( g \( x \) \) =x rSup { size 8{2} } - 4} {}

  • Let’s test it. f ( g ( x ) ) size 12{f \( g \( x \) \) } {} means do g size 12{g} {} , then f size 12{f} {} . What happens if we start with x = 9 size 12{x=9} {} ?

7 g ( x ) 7 + 2 = 9 f ( x ) ( 9 ) 2 4 ( 9 ) = 45 size 12{7 rightarrow g \( x \) rightarrow 7+2=9 rightarrow f \( x \) rightarrow \( 9 \) rSup { size 8{2} } - 4 \( 9 \) ="45"} {}

  • So, if it worked, our composite function should do all of that in one step.

7 x 2 4 = ( 7 ) 2 4 = 45 size 12{7 rightarrow x rSup { size 8{2} } - 4= \( 7 \) rSup { size 8{2} } - 4="45"} {} It worked!

There is a different notation that is sometimes used for composite functions. This book will consistently use f ( g ( x ) ) size 12{f \( g \( x \) \) } {} which very naturally conveys the idea of “plugging g ( x ) size 12{g \( x \) } {} into f ( x ) size 12{f \( x \) } {} .” However, you will sometimes see the same thing written as f ° g ( x ) size 12{f circ g \( x \) } {} , which more naturally conveys the idea of “doing one function, and then the other, in sequence.” The two notations mean the same thing.

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
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Lukman Reply
_3_2_1
felecia
⅗ ⅔½
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_½+⅔-¾
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
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Harshika Reply
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find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Mark
I need quadratic equation link to Alpa Beta
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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
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Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
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Opoku
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Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
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Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
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
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
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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?
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Source:  OpenStax, Precalculus with engineering applications. OpenStax CNX. Jan 24, 2011 Download for free at http://cnx.org/content/col11267/1.3
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