Section 1.8Combinations of Functions:

Composite Functions

1st Day

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows.

1. Sum: (f + g)(x) = f (x) + g(x)

2. Difference: (f − g)(x) = f (x) − g(x)

3. Product: (fg)(x) = f (x) ∙ g(x)

4. Quotient: , 0

f xfx g x

g g x

Example 1

Find (a) (f + g)(x), (b) (f – g)(x), (c) (fg)(x), and

(d) What is the domain of .fx

g

?f

g

1) f (x) = 2x + 5 and g(x) = 2 – x

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

= (2x + 5) + (2 – x)

= (2x – x) + (5 + 2)

= x + 7

b) (f − g)(x) = f (x) − g(x)

= (2x + 5) − (2 – x)

= (2x + x) + (5 − 2)

= 3x + 3

1) f (x) = 2x + 5 and g(x) = 2 – x

c) (f g)(x) = f (x) ∙ g(x)

= (2x + 5)(2 – x)

= 4x – 2x2 + 10 − 5x

= −2x2 − x + 10

d) , 0

f xfx g x

g g x

2 5

, 22

xx

x

2) f (x) = 3x – 2 and g(x) = x + 7

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

= (3x − 2) + (x + 7)

= (3x + x) + (−2 + 7)

= 4x + 5

b) (f − g)(x) = f (x) − g(x)

= (3x − 2) − (x + 7)

= (3x − x) + (−2 − 7)

= 2x − 9

2) f (x) = 3x − 2 and g(x) = x + 7

c) (f g)(x) = f (x) ∙ g(x)

= (3x − 2)(x + 7)

= 3x2 + 21x − 2x − 14

= 3x2 + 19x − 14

d) , 0

f xfx g x

g g x

3 2

, 77

xx

x

2

22

3) 4 and 1

xf x x g x

x

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

22

24

1

xx

x

b) (f − g)(x) = f (x) − g(x)

22

24

1

xx

x

2

22

3) 4 and 1

xf x x g x

x

c) (fg)(x) = f (x) ∙ g(x)

2

22

41

xx

x

2 2

2

4

1

x x

x

22

24

1

xx

x

d) , 0

f xfx g x

g g x

22

2

14

xx

x

2 2

2

1 4, 0

x xx

x

Example 2Evaluate the indicated function for f (x) = x2 +1 and g(x) = x – 4.

1) (f – g)(−1)

(f – g)(−1) = f (−1) − g(−1)

= [(−1)2 + 1] − [−1 − 4]

= 2 − (−5)

= 7

2) (fg)(5) + f (4)

(fg)(5) + f (4) = f (5) ∙ g(5) + f (4)

= [(5)2 + 1][5 − 4] + [(4)2 + 1]

= (26)(1) + 17

= 26 + 17

= 43

3) (f + g)(t – 2)

(f + g)(t – 2) = f (t − 2) + g(t − 2)

= [(t − 2)2 + 1] + [(t − 2) − 4]

= [t2 − 4t + 4 + 1] + [t − 6]

= t2 – 4t + 5 + t – 6

= t2 – 3t – 1

HW: p. 89 (5-23 odd)

2nd DayThe composition of the function f with the function g is:

(f ◦ g)(x) = f (g(x))

The domain of the composition is the set of all x in the domain of g.

Example 1

Given f (x) = x + 2 and g(x) = 4 – x2. Find:

a) (f ◦ g)(x)

b) (g ◦ f )(x)

c) (g ◦ f )(−2)

Given f (x) = x + 2 and g(x) = 4 – x2.

a) (f ◦ g)(x)

(f ◦ g)(x) = f (g(x))

= f (4 – x2)

= [4 – x2] + 2

= 6 – x2

1) Given f (x) = x + 2 and g(x) = 4 – x2.

b) (g ◦ f )(x)

(g ◦ f ) (x) = g (f (x))

= g(x + 2)

= 4 – (x + 2)2

= 4 – [x2 + 4x + 4]

= 4 – x2 – 4x – 4

= –x2 – 4x

c) (g ◦ f )(−2) = –(−2)2 – 4(−2)

= 4

Example 2

Given f (x) = 3x + 5 and g(x) = 5 – x. Find:

a) (f ◦ g)(x)

b) (g ◦ f )(x)

c) (f ◦ f )(x)

Given f (x) = 3x + 5 and g(x) = 5 – x.

a) (f ◦ g)(x)

(f ◦ g)(x) = f (g(x))

= f (5 – x)

= 3[5 – x] + 5

= 15 – 3x + 5

= 20 – 3x

Given f (x) = 3x + 5 and g(x) = 5 – x.

b) (g ◦ f )(x)

(g ◦ f ) (x) = g (f (x))

= g(3x + 5)

= 5 – (3x + 5)

= 4 – 3x – 5

=–3x – 1

Given f (x) = 3x + 5 and g(x) = 5 – x.

c) (f ◦ f )(x)

(f ◦ f ) (x) = f (f (x))

= f (3x + 5)

= 3(3x + 5) + 5

= 9x + 15 + 5

= 9x + 20

Example 3

Given: f (x) = x2 – 9 and

find the composition (f ◦ g)(x). Then find the domain of (f ◦ g)(x).

29 ,g x x

Given: f (x) = x2 – 9 and

(f ◦ g)(x) = f (g(x))

The domain of (f ◦ g)(x) is the domain of g(x),which is [−3, 3].

29 .g x x

29f x

229 9x

29 9x 2x

-4 -3 -2 -1 1 2 3 4

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

Example 4

Given: and g(x) = x + 1, find

(a) (f ◦ g)(x)

(b) (g ◦ f )(x) Find the domain of each function and each composite function.

2

3

1f x

x

Domain of f (x):

Domain of g(x):

, 1 1, 1 1,

,

Given: and g(x) = x + 1, find

(a) (f ◦ g)(x)

(f ◦ g)(x) = f (g(x))

= f (x + 1)

2

3

1f x

x

2

3

1 1x

2

3

2 1 1x x

2

3

2x x

, 2 2, 0 0, Domain of (f ◦ g)(x) :

Given: and g(x) = x + 1, find

(b) (g ◦ f )(x)

(g ◦ f )(x) = g( f (x))

Domain:

2

3

1f x

x

2

3

1g

x

2

31

1x

, 1 1, 1 1,

HW: pp. 89-90 (31-41 odd)