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 38E
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)=x3 with domain [3,+) , (gf)(x)=x23 with domain (,3][3,+) , (ff)(x)=x4 with domain (,+) and (ff)(x)=x4 with domain [12,+) .

Explanation of Solution

Given information:

The given functions are f(x)=x2 and g(x)=x3 .

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(x3)=(x3)2=x3

The domain of the function g(x) is [3,+) . So, the domain of the function is [3,+) .

The composite function (gf) can be calculated as:

(gf)(x)=g(f(x))=g(x2)=x23

The domain of the function f(x) is (,+) . But the composite function is not defined at x=±3 . So, the domain of the function (gf) is (,3][3,+) .

The composite function (ff) can be calculated as:

(ff)(x)=f(f(x))=f(x2)2=x4

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(x3)=x33

The domain of the function g(x) is [3,+) . The composite function is defined for x12 . So, the domain of the function (gg) is [12,+) .

Therefore, the functions are (fg)(x)=x3 with domain [3,+) , (gf)(x)=x23 with domain (,3][3,+) , (ff)(x)=x4 with domain (,+) and (ff)(x)=x4 with domain [12,+) .

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