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Chapter 5: Q. 36 (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)=1x+3,g(x)=-2x

Short Answer

Expert verified

a) f鈭�g=x3x-2, its domain is:x|x鈮�0,x鈮�23.

b)g鈭�f=-2(x+3), its domain is:x|x鈮�-3

c)localid="1648200188428" f鈭�f=x+33x+10,its domain is:localid="1648200197674" x|x鈮�-3,x鈮�-103

d)g鈭�g=x, its domain is:x|x鈮�0

Step by step solution

01

Step 1. Given information:

The functions are:

f(x)=1x+3,g(x)=-2x

f(x)is defined for x鈮�-3.

g(x) is defined forx鈮�0

02

Step 2. Find f∘g and its domain:

f鈭�g=f(g(x))=1-2x+3=1-2+3xx=x3x-2

This is defined when,

3x-2鈮�0x鈮�23

and

for the g(x)to be defined x鈮�0

So, Domain is:x|x鈮�23,x鈮�0

03

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

g鈭�f=g(f(x))=-21x+3=-2(x+3)

The domain of the function is the domain of f(x)

So, the domain is:x|x鈮�-3.

04

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

f鈭�f=f(f(x))=11x+3+3=11+3x+9x+3=x+33x+10

This is defined when

3x+10鈮�0x鈮�-103

The domain is:

x|x鈮�-103,x鈮�-3

05

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

g鈭�g=g(g(x))=-2-2x=x

The domain is;x|x鈮�0

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Most popular questions from this chapter

If f(7)=3 and f is one-to-one, what isf-1(13)?

Evaluate each expression using the values given in the table.

(a) f鈭�g1(b) f鈭�g2(c) g鈭�f2(d) g鈭�f3(e) g鈭�g1(f) f鈭�f3.

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

f(x)=2x3x-1

solve each equation. Verify your results using a graphing utility.

5x2+8=1252x

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

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