7.4 Composition of Functions

3/26/12

Function:

A function is a special relationship between values: Each of its input values gives back exactly one output value.

It is often written as "f(x)" where x is the input value.

Ex 1: Gfind: a.) Solution: Substitute 0 for x in 3x+ 1

3(0) + 1 = 1 = 1

b.) 3(½) + 1 = 1 + 1 = 2

c.) 3(a) + 1 = 3a + 1

d.) 3(a+3) + 1 = 3a + 9 +1 = 3a+10

e.) = 3a + 4

Ex 2: Gfind: a.)

= -5

b.)

= -c.)

=

d.)

= e.)

3 =

f

g

y

x

2

4

6

8

2 4 6 8–2–4–6–8

–2

–4

Use the graph to find:a. f(0) Solution: at x = 0, the graph of the f (red) function is at 2.f(0) = 2b. g(0)Solution: at x = 0, the graph of the g (black) function is at 3g(0) = 3

Composition of functionsoccurs when you insert one function into another. In effect, the range (output) of the inside function becomes the domain (input) of the outside function. The notation for composition of functions is either

Or

Note: which means order matters!

f

g

y

x

2

4

6

8

2 4 6 8–2–4–6–8

–2

–4

Use the graph to find:c.) f(g(0))Solution: Evaluate g(0) first, and from the previous problem (b), g(0) = 3. Then look at the graph of the f function and see what the y component when x = 3.At x = 3 the red graph is at -1f(g(0)) = -1

f

g

y

x

2

4

6

8

2 4 6 8–2–4–6–8

–2

–4

Use the graph to find:d.) g(f(0))Solution: Evaluate f(0) which in problem (a) is 2. Then look at the graph of the g function and see what the y component when x = 2.At x = 2 the black graph is at 5.g(f(0)) = 5

f

g

y

x

2

4

6

8

2 4 6 8–2–4–6–8

–2

–4

Use the graph to find:a.)(f ⃘g)(-1) and (g ⃘f)(-1) a.) Solution: Evaluate g(-1)g(-1) = 2 then evaluate f(2)f(2) = 0b.) Solution: Evaluate f(-1)f(-1) = 3 then evaluate g(3)g(3) = 6

Given:

Evaluate the following expressions:a. f(-1)b. g(-1)c. f(g(-1))d. g(f(-1))

a.)

b.)

c.)

d.)

Example 1 Add and Subtract Functions

Let and . Find: =f ( )x 4x 2 =g( )x x 1+

f ( )x g( )x+a. b. =h( )x f ( )x g( )x–

SOLUTION

=h( )x f ( )x g( )x+a. =h( )x f ( )x g( )xb. –

4x 2 ( )1x ++= 4x 2 ( )1x += –

In both parts and , the domains of f and g are all real numbers. So, the domain of h is all real numbers.

( )a ( )b

4x 2 1x ++= 4x 2 1x= – –

Let and . Find:

Example 2 Multiply and Divide Functions

=f ( )x x 3 =g( )x 2x

f ( )x g( )xa. • b. =h( )xf ( )x

g( )x

=2x 4

=21

x 2

𝑓 (𝑥 ) ∙𝑔 (𝑥 )=𝑥3 ∙2𝑥 𝑓 (𝑥 )𝑔 (𝑥 )

= 𝑥3

2𝑥

Checkpoint Perform Function Operations

1.

Let and . Find=f ( )x 3x =g( )x x 1–

f ( )x g( )x+ANSWER

4x 1–

3x 2 3x,–2. f ( )x g( )x•

3. f ( )x

g( )x

3x

x 1–, x 1=

Composition of

Functions:

is the process of combining two functions where one function is substituted in place of each x in the other function.

Vocabulary

f(g(x)) read as the “composition of f with g”.g(f(x)) read as the “composition of g with f”.

In Symbols:

Important Note:

f(g(x)) is not the same as g(f(x)).The order of functions when they are composed is very important.

Example 3 Write a Composition of Functions

=f ( )x x 2 =g( )x 2x 3+Let and . Find the following.

g( )x( )fa. f ( )x( )gb.

SOLUTION

Write the composition by substituting the expression for the inner function in the outer function, and simplify.

g( )x( )fa. = ( )3+2xf = ( )23+2x = 4x 2 12x + 9+

f ( )x( )gb. = g( )x 2 = 2( )x 2 3+ = 2x 2 3+

Example 4 Evaluate a Composition of Functions

=g( )x 5x=f ( )x x 2 3+Let and . Evaluate .g( )2( )f

SOLUTION

To evaluate , first find :g( )2( )f g( )2

g( )2 5( )2= = 10

g( )2 = 10Then substitute into : g( )2( )f

g( )2( )f = f ( )10 = 102 3+ = 100 3+ = 103

Checkpoint Find and Evaluate Compositions of Functions

Let and . Find the composition. Then evaluate the composition when x 2.

=f ( )x x 2 =g( )x x 1–=

4. g( )x( )f

5. f ( )x( )g

ANSWER x 2 2x + 1; 1–

ANSWER x 2 1; 3–

Example 5 Model a Real-World Situation

Hair Salon You have a coupon for $10 off the cost of your purchase at a hair salon. The salon also offers a discount off your purchase, as shown.

=f ( )x x 10–

=g( )x 0.85x

f ( )x( )g

Let x be the cost of your purchase. Then is the cost of the purchase using your coupon, and is the cost of your purchase with the salon’s discount. Find . Tell what it represents.

Example 5 Model a Real-World Situation

SOLUTION

f ( )x( )g = ( )10xg =– ( )10x0.85 – = 8.50.85x –

The composition represents the cost of your purchase when the $10 coupon is applied before the 15% discount.

f ( )x( )g

Homework:7.4 p.376 #14-24even, 26, 2732-38even, 40-46 all