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Chapter 5: Q. 40 (page 257)

In Problems 29 鈥� 44, for the given functions f and g, find:

(a)f鈭�g

(b)g鈭�f

(c) f鈭�f

(d) g鈭�g

State the domain of each composite function.

f(x)=x-2g(x)=1-2x

Short Answer

Expert verified

a) f鈭�g=-1-2xand its domain is:x鈮�-12

b) g鈭�f=1-2x-2, and its domain is:x鈮�2

c) f鈭�f=x-2-2and its domain is:x鈮�6.

d) g鈭�g=4x-1and its domain is:-鈭�,鈭�

Step by step solution

01

Step 1. Given information:

The functions are:

f(x)=x-2g(x)=1-2x

The domain of f(x)=x鈮�2

The domain ofg(x)=-鈭�,鈭�

02

Part (a) Step 1. Find f∘g and its domain.

f鈭�g=fg(x))=f(1-2x)=1-2x-2=-1-2x

This is defined when g(x)is defined and

-1-2x鈮�0x鈮�-12

So domain is:x鈮�-12

03

Part (b) Step 1. Find g∘f and its domain.

g鈭�f=g(f(x))=g(x-2)=1-2x-2

This is defined only when f(x)is defined and x-2鈮�0x鈮�2

So the domain is:x鈮�2

04

Part (c) Step 1. Find f∘f and its domain.

f鈭�f=f(f(x))=f(x-2)=x-2-2

This is defined whenf(x)is defined and

x-2-2鈮�0x-2鈮�2x-2鈮�4x鈮�4+2x鈮�6

So domain is:

x鈮�6

05

Part (d) Step 1. Find g∘g and its domain.

g鈭�g=g(g(x))=g(1-2x)=1-2(1-2x)=1-2+4x=4x-1

This is defined when g(x)is defined.

The domain of this is-鈭�,鈭�

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