Function Operations 8.58.5

1. Add or subtract functions.2. Multiply functions.

Composite Functions12.112.11. Find the composition of two functions.

Add the following polynomials.

5x + 1 3x2 – 7x + 6

3x2 – 2x + 7

f(x) = 5x + 1

(f + g)(x) = = (5x + 1) + (3x2 – 7x + 6) = 3x2 – 2x + 7

g(x) = 3x2 – 7x + 6

f(x) + g(x)Always rewrite!!!

Copyright © 2011 Pearson Education, Inc.

Adding or Subtracting Functions

(f + g)(x) = f(x) + g(x)

(f – g)(x) = f(x) – g(x).

f(x) = 3x + 1 g(x) = 5x + 2Find:

(f + g)(x) (f - g)(x)

(g - f)(x) (f - g)(-2)

= f(x) + g(x)

= (3x + 1) + (5x + 2)

= 8x + 3

= f(x) – g(x)

= (3x + 1) – (5x + 2)

= -2x – 1

= 3x + 1 – 5x – 2

= g(x) – f(x)

= (5x + 2) – (3x + 1)

= 2x + 1

= 5x + 2 – 3x – 1

= f(-2) – g(-2)

f(-2) = 3(-2) + 1 = -5

= (-5) – (-8 )

g(-2)= 5(-2) + 2 = -8

= 3

Always rewrite!!!

Slide 3- 5Copyright © 2011 Pearson Education, Inc.

Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)?

a) x + 4

b) x − 4

c) 9x + 1

d) 9x – 1

8.5

Slide 3- 6Copyright © 2011 Pearson Education, Inc.

Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)?

a) x + 4

b) x − 4

c) 9x + 1

d) 9x – 1

8.5

Multiplying Functions

(f g)(x) = f(x)∙g(x).

Cop

yright ©

20

11 P

earso

n E

du

catio

n, Inc.

f(x) = 2x + 7 and g(x) = x − 4

Find (f g)(x).

= (2x + 7)(x − 4)

= 2x2 − 8x + 7x – 28

= 2x2 − x – 28

(f g)(x) = f(x)∙g(x)

Always rewrite!!!

f(x) = – x2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find:

(gh)(x) (fg)(0)

(fh)(-1) (f h)(x)

=g(x) ∙ h(x)

= (x + 2)(x – 8)

= x 2 – 6x - 16

= f(0) ∙ g(0)

= (2)(2)

= 4

= f(-1) ∙ h(-1)

= (9)(-9)

= -81

= f(x) ∙ h(x)

= (-x2 – 8x + 2)(x – 8)

= -x 3 + 66x – 16

f(-1) = -(-1) 2 – 8(-1) + 2 = -1 + 8 + 2 = 9

Slide 3- 10Copyright © 2011 Pearson Education, Inc.

Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f g)(x)?

a) 15x2 − 13x + 2

b) 15x2 − 13x − 2

c) 15x2 − 7x + 2

d) 15x2 − 7x − 2

8.5

Slide 3- 11Copyright © 2011 Pearson Education, Inc.

Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f • g)(x)?

a) 15x2 − 13x + 2

b) 15x2 − 13x − 2

c) 15x2 − 7x + 2

d) 15x2 − 7x − 2

8.5

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

f (2) =

f (a) =

f (x+4) =

f (g(x)) =

2(2) + 3 = 7

2a + 3

2(x + 4) + 3 =

Composition of Functions

(f ◦ g)(x) =

Nested FormatNested Format

2x + 8 + 3 = 2x + 11

Composition of Functions

f g x f g x

g f x g f x

Shorthand notation for substitution.Shorthand notation for substitution.

Nested FormatNested Format

Always rewrite composition of functions in nested format!Always rewrite composition of functions in nested format!

Read “f of g of x”.Read “f of g of x”.

If and find .( ) 3 8f x x ( ) 2 5,g x x 3f g

3 3f g f g

1f

3 81

11

Find f(1).

Simplify.

Substitute 1 for g(3)

Find g(3).

Write in nested format.

g(3) = 2(3) – 5 = 1

f(x) = x2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find: 3hg xfh

xgf

= g(h(3))

h(3) = 3 – 8 = -5

= g(-5)

= -3

= h(f(x))

= h(x2 – 8x + 2)

= (x2 – 8x + 2) - 8

= f(g(x))

= f(x + 2)

= (x + 2)2 – 8(x + 2) + 2

= x2 – 8x – 6

= x2 + 4x + 4 – 8x – 16 + 2

= x2 – 4x – 10

(x + 2)2 (x + 2)(x + 2)x2 + 4x + 4

(x + 2)2 x2 + 4X

g(-5) = -5 + 2 = -3

Rewrite & Foil

Always rewrite!!!

Slide 12- 16Copyright © 2011 Pearson Education, Inc.

If f(x) = x + 7 and g(x) = 2x – 12, what is

a) 44

b) 3

c) 3

d) 44

4 .f g

Slide 12- 17Copyright © 2011 Pearson Education, Inc.

If f(x) = x + 7 and g(x) = 2x – 12, what is

a) 44

b) 3

c) 3

d) 44

4 .f g