BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.6, Problem 39E
To determine

To find: The functions fg , gf , ff and gg , and their domains.

Expert Solution

Answer to Problem 39E

The functions are (fg)(x)=|2x+3| with domain (,+) , (gf)(x)=2|x|+3 with domain (,+) , (ff)(x)=x with domain (,+) and (gg)(x)=4x+9 with domain (,+) .

Explanation of Solution

Given information:

The given functions are f(x)=|x| and g(x)=2x+3 .

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+3)=|2x+3|

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)2=g(f(x))=g(|x|)=2|x|+3

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(|x|)=||x||

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(2x+3)=2(2x+3)+3=4x+9

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

Therefore, the functions are (fg)(x)=|2x+3| with domain (,+) , (gf)(x)=2|x|+3 with domain (,+) , (ff)(x)=x with domain (,+) and (gg)(x)=4x+9 with domain (,+) .

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