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 37E
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)=12x+4 with domain (,2)(2,+) , (gf)(x)=2x+4 with domain (,0)(0,+) , (ff)(x)=x with domain (,+) and (gg)(x)=4x+12 with domain (,+) .

Explanation of Solution

Given information:

The given functions are f(x)=1x and g(x)=2x+4 .

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+4)=12x+4

The domain of the function g(x) is (,+) . But the composite function is not defined at x=2 . So, the domain of the function is (,2)(2,+) .

The composite function (gf) can be calculated as:

(gf)(x)=g(f(x))=g(1x)=2x+4

The domain of the function f(x) is (,0)(0,+) . So, the domain of the function (gf) is (,0)(0,+) .

The composite function (ff) can be calculated as:

(ff)(x)=f(f(x))=f1x=11x=x

The domain of the function f(x) is (,0)(0,+) . But the above function is defined for all real numbers. So, the domain of the function (ff) is (,+) .

The composite function (gg) can be calculated as:

(gg)(x)=g(g(x))=g(2x+4)=2(2x+4)+4=4x+12

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

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

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