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
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