1.8 Combinations of

Functions

JMerrill, 2010

Arithmetic Combinations

Sum

Let

Find (f + g)(x)

( )( ) ( ) ( )f g x f x g x 2 2( ) 5 2 3 ( ) 4 7 5 f x x x g x x x

2 2

2

(5 2 3) (4 7 5)

9 5 2

x x x x

x x

Difference

Let

Find (f - g)(x)

( )( ) ( ) ( )f g x f x g x 2 2( ) 5 2 3 ( ) 4 7 5 f x x x g x x x

2 2

2

(5 2 3) (4 7 5)

9 8

x x x x

x x

Product

Let

Find

( )( ) ( ) ( ) fg x f x g x2( ) 5 ( ) 3 1 f x x g x x

( )( )fg x

2

3 2

5 (3 1)

15 5

x x

x x

Quotient

Let

Find

( )( ) ( ) 0

( )

f f xx where g x

g g x

2( ) 5 ( ) 3 1 f x x g x x

( )f

xg

25

3 1

x

x

You Do: Let Find:

(f+g)(x)(f•g)(x)

(f-g)(x)

(g-f)(x)

2( ) 3, ( ) 9 f x x g x x

( )f

xg

2 12x x

2 6x x

2 6x x

3 23 9 27x x x

1

3x

Finding the Domain of Quotients of Functions To find the domain of the quotient,

first you must find the domain of each function. The domain of the quotient is the overlap of the domains.

Example

The domain of f(x) = The domain of g(x) = [-2,2]

2Given f(x) x and g(x) 4 x

gfFind the domains of (x) and (x)

g f

[0, )

Example

Since the domains are: f(x) = g(x) = [-2,2] The domains of the quotients are

2

ff (x) xx

g g(x) 4 x

[0, )

2g g(x) 4 x(x)

ff (x) x

gf:[0,2) : (0,2]

g f

Composition of Functions Most situations are not modeled by

simple linear equations. Some are based on a system of functions, others are based on a composition of functions.

A composition of functions is when the output of one function depends on the input from another function.

Compositions Con’t For example, the amount you pay on

your income tax depends on the amount of adjusted gross income (on your Form 1040), which, in turn, depends on your annual earnings.

Composition Example In chemistry, the process to convert

Fahrenheit temperatures to Kelvin units

This 2-step process that uses the output of the first function as the input of the second function.

5( ) ( 32)

9 c t f

This formula gives the Celsius temp. that corresponds to the Fahrenheit temp.

( ) 273k c c This formula converts the Celsius temp. to Kelvins

Composition Notation (f o g)(x) means f(g(x))

(g o f)(x) means g(f(x)

Composition of Functions: A Graphing Approach

( )( 1)Find f g ( ( 1))f g (3)f(3) 0f

( )(3)Find g f( (3))g f(0)g(0) 2g

(f g)(x) and (g f )(x)

You Do f(g(0)) = g(f(0)) =

(f°g)(3) = (f°g)(-3) =

(g°f)(4) = (f°g)(4) =

f(x)

g(x)

4

4

3

3

0

0

Compositions: Algebraically Given f(x) = 3x2 and g(x) = 5x+1 Find f(g(2)) Find g(f(4)) g(2)=5(2)+1 = 11 f(11) = 3(11)2

=363

How much is f(4)?

g(48) = 5(48)+1=241

Compositions: Algebraically Con’t Given f(x) = 3x2 and g(x) = 5x+1 Find f(g(x)) Find g(f(x)) What does g(x)=? f(5x+1) =3(5x+1)2

=3(25x2+10x+1) =75x2+30x+3

What does f(x)=?

g(3x2) = 5(3x2)+1=15x2+1

You Do f(x)=4x2-1 g(x) = 3x

Find: (f g)(x) (g f )(x)

2

2

2

f (g(x))

f(3x)

4(3x) 1

4(9x ) 1

36x 1

2

2

2

g(f(x))

g(4x 1)

12x 3

3(4x 1)