Section 5.1

Composite Functions

Suppose that an oil tanker is leaking oil and we want to be able to determine the area o the circular oil patch around the ship. It is determined that the oil is leaking from the tanker in such a way that the radius of the circular oil patch around the ship is increasing at a rate of 3 feet per minute.

2 3Suppose that 2 3 4 1. Find:f x x g x x

(a) 1 1f g f g

31 4 1 1 5g

235 2 35 5f

(b) 1 1g f g f

21 2 1 3 5f

35 54 1 501g

(c) 2 2f ff f

22 2 2 3 11f

211 112 3 245f

(d) 1 1gg g g

31 4 1 1 3g

31 73 4 103g

2Suppose that 2 4 and 4 1. f x x x g x x

(a) f g x xf g

4 1g x x

224 1 4 1 41 4x x xf

22 16 8 1 4 1 4x x x 232 16 2 4 3x x x

232 12 5x x

The domain of g is all real numbers as is the domain of the composite function, so the domain of f ◦ g is the set of all real numbers.

2Suppose that 2 4 and 4 1. f x x x g x x

(b) g f x xg f

22 4f x x x

2 242 4 2 4 1g x x x x

28 4 16 1x x

The domain of f is all real numbers as is the domain of the composite function, so the domain of g ◦ f is the set of all real numbers.

28 4 17x x

3 2 and .

5 1f x g x

x x

We first need the domain of which is 1 .g x x

So we must exclude 1 from the domain of .f g

The domain of is 5 so if we put in this

function for , 5.

f x g x

x g x

2Find value where 5

1g x

x

2 5 1 x 2 5 5x 3

5x 3

The domain of is 1, .5

f g x x x

3 2 Suppose that and .

5 1Find: (a) (b) and the domain of the composite function.

f x g xx x

f g f f

Note the domain of is 5 and the domain of is 1 .f x x g x x

(a) f g x xf g

2

1g x

x

232

151x

x

f

3 1 3 3 3 3 3 3

2 5 1 2 5 5 5 3 5 3

x x x x

x x x x

The domain of is the excluded value for the domain of which is 1and also the value that would cause the composite function to have division by 0

3so the domain of is 1, .

5

f g g

f g x x x

3 2 Suppose that and .

5 1Find: (a) (b) and the domain of the composite function.

f x g xx x

f g f f

Note the domain of is 5 and the domain of is 1 .f x x g x x

(b) f f x xf f

3

5f x

x

333

555x

x

f

3 5 3 15 3 15 3 15

3 5 5 3 5 25 5 28 5 28

x x x x

x x x x

The domain of is the excluded value for the domain of which is 5and also the value that would cause the composite function to have division by 0

28so the domain of is 5, .

5

f f f

f f x x x

1If 2 1 and 1 . Show that =

2for every in the domain of and .

f x x g x x f g x g f x x

x f g g f

xf g x f g

11

2g x x

1 11

2 22 11xf x

1 12 1 1 1

2 2x x x

xg f x g f

2 1f x x

12 1 2 1 1

2x xg

12

2x x

50f x x 2 1g x x

2 1x

2

3Find functions and such that if

5f g f g H H x

x

5g x x

5x

2

3f x

x

2

3=

5g x

xf g x H xf