# The functions f ∘ g , g ∘ f , f ∘ f and g ∘ g , and their domains. ### 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 39E
To determine

## To find: The functions f∘g , g∘f , f∘f and g∘g , and their domains.

Expert Solution

The functions are (fg)(x)=|2x+3| with domain (,+) , (gf)(x)=2|x|+3 with domain (,+) , (ff)(x)=x with domain (,+) and (gg)(x)=4x+9 with domain (,+) .

### Explanation of Solution

Given information:

The given functions are f(x)=|x| and g(x)=2x+3 .

Calculation:

The domain of a function is the set of all numbers for which the given function is defined.

The composite function can be calculated as:

(fg)(x)=f(g(x))=f(2x+3)=|2x+3|

The domain of the function g(x) is (,+) . So, the domain of the function is (,+) .

The composite function (gf) can be calculated as:

(gf)(x)2=g(f(x))=g(|x|)=2|x|+3

The domain of the function f(x) is (,+) . So, the domain of the function (gf) is (,+) .

The composite function (ff) can be calculated as:

(ff)(x)=f(f(x))=f(|x|)=||x||

The domain of the function f(x) is (,+) . So, the domain of the function (ff) is (,+) .

The composite function (gg) can be calculated as:

(gg)(x)=g(g(x))=g(2x+3)=2(2x+3)+3=4x+9

The domain of the function g(x) is (,+) . So, the domain of the function (gg) is (,+) .

Therefore, the functions are (fg)(x)=|2x+3| with domain (,+) , (gf)(x)=2|x|+3 with domain (,+) , (ff)(x)=x with domain (,+) and (gg)(x)=4x+9 with domain (,+) .

### 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!