# 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 41E
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)=2x12x with domain (,0)(0,+) , (gf)(x)=x1x+1 with domain (,1)(1,+) , (ff)(x)=x2x+1 with domain (,1)(1,12)(12,+) and (gg)(x)=4x3 with domain (,+) .

### Explanation of Solution

Given information:

The given functions are f(x)=xx+1 and g(x)=2x1 .

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(2x1)=2x12x1+1=2x12x

The domain of the function g(x) is (,+) . But the composite function is not defined at x=0 . So, the domain of the function is (,0)(0,+) .

The composite function (gf) can be calculated as:

(gf)(x)=g(f(x))=g(xx+1)=2(xx+1)1=x1x+1

The domain of the function f(x) is (,1)(1,+) . So, the domain of the function (gf) is (,1)(1,+) .

The composite function (ff) can be calculated as:

(ff)(x)=f(f(x))=f(xx+1)=xx+1xx+1+1=x2x+1

The domain of the function f(x) is (,1)(1,+) . The composite function is not defined at x=12 . So, the domain of the function (ff) is (,1)(1,12)(12,+) .

The composite function (gg) can be calculated as:

(gg)(x)=g(g(x))=g(2x1)=2(2x1)1=4x3

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

Therefore, the functions are (fg)(x)=2x12x with domain (,0)(0,+) , (gf)(x)=x1x+1 with domain (,1)(1,+) , (ff)(x)=x2x+1 with domain (,1)(1,12)(12,+) and (gg)(x)=4x3 with domain (,+) .

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