Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Question

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 & Answer

Answer to Problem 37E

The functions are (f∘g)(x)=12x+4 with domain (−∞,−2)∪(−2,+∞) , (g∘f)(x)=2x+4 with domain (−∞,0)∪(0,+∞) , (f∘f)(x)=x with domain (−∞,+∞) and (g∘g)(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:

(f∘g)(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 (g∘f) can be calculated as:

(g∘f)(x)=g(f(x))=g(1x)=2x+4

The domain of the function f(x) is (−∞,0)∪(0,+∞) . So, the domain of the function (g∘f) is (−∞,0)∪(0,+∞) .

The composite function (f∘f) can be calculated as:

(f∘f)(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 (f∘f) is (−∞,+∞) .

The composite function (g∘g) can be calculated as:

(g∘g)(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 (g∘g) is (−∞,+∞) .

Therefore, the functions are (f∘g)(x)=12x+4 with domain (−∞,−2)∪(−2,+∞) , (g∘f)(x)=2x+4 with domain (−∞,0)∪(0,+∞) , (f∘f)(x)=x with domain (−∞,+∞) and (g∘g)(x)=4x+12 with domain (−∞,+∞) .

Chapter 2.6, Problem 36E

Chapter 2.6, Problem 38E

Chapter 2 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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