# 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 36E
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)=x+2 with domain (,+) , (gf)(x)=x3+23 with domain (,+) , (ff)(x)=x9+6x6+12x3+10 with domain (,+) and (gg)(x)=x33 with domain (,+) .

### Explanation of Solution

Given information:

The given functions are f(x)=x3+2 and g(x)=x3 .

Calculation:

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

The composite function can be calculated as:

(fg)(x)=f(g(x))=f(x3)=(x3)3+2=x+2

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)=g(f(x))=g(x3+2)=x3+23

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(x3+2)=(x3+2)3+2=x9+6x6+12x3+10

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(g(x))=g(x3)=x33

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

Therefore, the functions are (fg)(x)=x+2 with domain (,+) , (gf)(x)=x3+23 with domain (,+) , (ff)(x)=x9+6x6+12x3+10 with domain (,+) and (gg)(x)=x33 with domain (,+) .

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