Section 5.1 Composite Functions
description
Transcript of Section 5.1 Composite Functions
Section 5.1Composite 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
2 35 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
3 1 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 1
f x g xx 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 xx g x
2Find value where 5 1
g xx
2 5 1 x 2 5 5x
35
x 3The domain of is 1, .5
f g x x x
3 2 Suppose that and . 5 1
Find: (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
21
g xx
232
151x
x
f
3 1 3 3 3 3 3 32 5 1 2 5 5 5 3 5 3
x x x xx 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 1
Find: (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
35
f xx
333
555x
x
f
3 5 3 15 3 15 3 153 5 5 3 5 25 5 28 5 28
x x x xx 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 = 2
for 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
1 12
g x x
1 112 2
2 11xf x
1 12 1 1 12 2
x x x
xg f x g f
2 1f x x
12 1 2 1 12
x xg
1 22
x x
50f x x 2 1g x x
2 1x
2
3Find functions and such that if 5
f g f g H H xx
5g x x
5x
2
3f xx
2
3= 5
g xx
f g x H xf