Composition of Functions
| Site: | Clare-Gladwin RESD |
| Course: | Michigan Algebra II KHauck |
| Book: | Composition of Functions |
| Printed by: | Guest user |
| Date: | Friday, November 22, 2024, 5:05 PM |
Description
Introduction
Composition of functions is the process of using one function rule as the input in a second function rule. A composite function is a function that represents in one function, the result of an entire chain of dependent functions. Since such chains are very common in real life, finding composite functions is a very important skill.Evaluating
Suppose you are given the two functions f(x) = 2x + 3 and g(x) = –x2 + 5. Composition means that you can substitute g(x) as the input of f(x).
This is written as (f o g)(x) or f(g(x)). The value of x is substituted as the input of g(x) and simplified. The result is then substituted as the input of f(x) and simplified again.
Example Given f (x) = 2x + 3 and g (x) = –x2 + 5 , find (g o f)(1) .
(g o f)(1) = g( f(1))
Step 1. Substitute the value of x as the input of the innermost function first, in this case f(x).
f (x) = 2x + 3
f (1) = 2(1) + 3
f (1) = 2 + 3
f (1) = 5
Step 2. Use this output as the input of the next innermost function, in this case g(x).
g (5) = -(5)2 + 5
g (5) = -25 + 5
g (5) = -20
Therefore, g(f(1)) = -20
Video Lesson
To learn how to evaluate composite functions at a given domain value, select the following link:
Symbolic Composition
Composition of functions can also be evaluated symbolically. To evaluate a symbolic composition, first substitute a variable as the input of the innermost function and then use that output as the input of the next function. In this case, it is not the intent to find a certain numerical value. Instead, the composite function is the formula that results from substituting the formula for g(x) into the formula for f( x).Example 1 Given f(x) = 2x + 3 and g(x) = -x2 + 5 , find (f o g)(x) .
(f o g)(x) = f(g(x))
= f(-x2 + 5)
Step 1. Rewrite the outer-function, leaving empty parentheses for the input expression.
f( ) = 2( ) + 3 ... setting up to insert the input formula
Step 2. Insert the function g(x) into the parentheses as the input expression.
f(-x 2 +5)= 2(-x2 + 5) + 3
Step 3. Simplify the right side of the equals sign.
f (-x2 + 5)= -2x2 + 10 + 3
f(g(x))= -2x2 + 13
Example 2
Given f(x) = 2x + 3 and g(x) = –x2 + 5 , find (g o f)(x) .
(g o f)(x) = g(f(x))
= g(2x + 3)
Step 1. Rewrite the outer-function, leaving empty parentheses for the input expression.
g( )= -1( )2 + 5 ... setting up to insert the input formula
Step 2. Insert the function f(x) into the parentheses as the input expression.
g(2x +3)= -1(2x + 3)2 + 5
Step 3. Simplify the right side of the equals sign.
g(2x + 3)= –1(4x2 +12x + 9) + 5
g(f(x))= –4x2 – 12x – 4
Video Lesson
To learn how to evaluate composite functions, select the following link:Writing Composite Functions
Application
You work forty hours a week at a furniture store. You receive a $220 weekly salary, plus a 3% commission on sales over $5000. Assume that you sell enough this week to get the commission.
Given the function f(x) = 0.03x and g(x) = x – 5000, which of (f o g)(x) and (g o f)(x) represents your commission?
Solution:
Using (f o g)(x) = f(g(x)) means subtract the sales x from $5000 that didn't earn commission, and then multiplying by 3%.
On the other hand, (g o f)(x) = g(f(x)) means multiply the sales x by 3%, and then subtract $5000 from the result. This could result in a negative number
Therefore, (f o g)(x) represents the commission.
Check the composite function with numbers that make sense in the situation. In the case of the commission function above, you could test the following sales values:
|
total sales |
$3000 |
$6000 |
$8000 |
|
commission |
$3000 – $5000 |
$6000 – $5000 |
$8000 – $5000 |
|
commission |
$0 |
(0.03)($1000) |
(0.03)($3000) |
For each sales value, subtract $5000 to see if there is a commission. Then multiply by 3%. The composite function would be f(g(x)) = 0.03(x - 5000).
Composition vs. Multiplication
Notice in the two previous symbolic examples, the resulting composite functions are not the same.
( f o g)(x) = –2x2 + 13
(g o f )(x) = –4x2 – 12x – 4
In general (f o g)(x) is not the same as (g o f )(x). In particular, composition is not the same as multiplication. The open dot is not the same as a multiplication dot, nor does it mean the same thing. While the following is true:
f (x) • g(x) = g(x) • f(x) [Commutative Property of Multiplication]
...you cannot say that:
(f o g)(x) = (g o f)(x) [generally false for composition]
Composition is not commutative like multiplication, and is an entirely different process.
Guided Practice
To solidify your understanding of function notation, operations and composition, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Interactive Activity
How to evaluate and express a composition of two functions.
Practice
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Video Lesson
Answer Key
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Sources
Sources used in this book: Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project Holt, Rinehard & Winston, "Properties and Attributes of Functions." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch09_04_homeworkhelp.html (accessed 6/29/2010). Kenny Felder, "Function Concepts -- Composite Functions," Connexions, December 29, 2008, http://cnx.org/content/m18187/1.2/. Math Warehouse, "Composition of Functions." http://www.mathwarehouse.com/algebra/relation/composition-of-function.php (accessed 08/09/2010). Purple Math, "Composition of Function." http://www.purplemath.com/modules/fcncomp4.htm (accessed 6/28/2010).