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Chapter 5: Q. 39 (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)=xg(x)=2x+3

Short Answer

Expert verified

a) f鈭�g=2x+3and its domain is:x鈮�-32

b)g鈭�f=2x+3 , and its domain:x鈮�0

c) f鈭�f=x4and its domain is:x鈮�0.

d) g鈭�g=4x+9and its domain is:-鈭�,鈭�

Step by step solution

01

Step 1. Given information:

The functions are:

f(x)=xg(x)=2x+3

The domain of f(x)={x|x鈮�0}

The domain ofg(x)is set of all real numbers.

02

Part (a)  Step 1. To find f∘g and its domain:  

f鈭�g=f(g(x))=2x+3

It is only defined when g(x)is defined and

2x+3鈮�0x鈮�-32

So its domain is:x鈮�-32

03

Part (a)  Step 1. To find g∘fand its domain:  

g鈭�f=g(f(x))=2x+3

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

So its domain is:x鈮�0

04

Part (c)  Step 1. To find f∘fand its domain:  

f鈭�f=f(x)=x=x4

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

So domain is:[0,鈭�)

05

Part (d)  Step 1. To find g∘gand its domain:  

g鈭�g鈭�=g(g(x))=g(2x+3)=2(2x+3)+3=4x+6+3=4x+9

This is defined for the set of all real values of x.

So domain is:(-鈭�,鈭�)

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