the function f ∘ g , g ∘ f , f ∘ f and g ∘ g and their domain

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.6, Problem 33E
To determine

To find : the function f∘g,g∘f,f∘f and g∘g and their domain

Expert Solution

(fg)(x)=8x+1;Dfg=(,+)(gf)(x)=8x+11;Dgf=(,+)(ff)(x)=4x+9;Dff=(,+)(gg)(x)=16x5;Dgg=(,+)

Explanation of Solution

Given information : The function is f(x)=2x+3&g(x)=4x1

Calculation : f(x)=2x+3,g(x)=4x1

The function f and g are linear functions so they are defined for every value of x in the set of real number

The domain of f is Df=(,+)

The domain of g is Dg=(,+)

For the function of fg

(fg)(x)=f(g(x))(fg)(x)=f(4x1)(fg)(x)=2(4x1)+3(fg)(x)=8x2+3(fg)(x)=8x+1

For the domain of this function:

D={xDgg(x)Df={x4x1={x

The domain of the composite function fg is Dfg=(,+)

For the function of gf

(gf)(x)=g(f(x))(gf)(x)=g(2x3)(gf)(x)=4(2x3)1(gf)(x)=8x+121(gf)(x)=8x+11

For the domain of this function:

D={xDff(x)Dg={x2x+3={x

The domain of the composite function gf is Dgf=(,+)

For the function of ff

(ff)(x)=f(f(x))(ff)(x)=f(2x3)(ff)(x)=2(2x3)+3(ff)(x)=4x+6+3(ff)(x)=4x+9

For the domain of this function:

D={xDff(x)Df={x2x+3={x

The domain of the composite function ff is Dff=(,+)

For the function of gg

(gg)(x)=g(g(x))(gg)(x)=g(4x1)(gg)(x)=4(4x1)1(gg)(x)=16x41(gg)(x)=16x5

For the domain of this function:

D={xDgg(x)Dg={x4x1={x

The domain of the composite function gg is Dgg=(,+)

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!