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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If f and g are both even functions, is the product of fg even? If f and g are both odd functions, is fg odd? What if f is even and g is odd? Justify your answers.

To determine

To determine and justify: Whether the function fg is even if f and g are even; whether the function fg is odd if f and g are odd; whether the function fg is even or odd if f  is even and g is odd.

Explanation

Definition used:

If f(x)=f(x), the function f(x) is said to be an even function.

If f(x)f(x), the function f(x) is not an even function.

If f(x)=f(x), the function f(x) is said to be an odd function.

If f(x)f(x), the function f(x) is not an odd function.

Calculation:

If f and g are even functions, then by the definition, f(x)=f(x) and g(x)=g(x)

Recall the fact that, (fg)(x)=f(x)g(x) (1)

As f and g are even functions, substitute f(x)=f(x) and g(x)=g(x) in equation (1) as follows.

(fg)(x)=f(x)g(x)=f(x)g(x)=(fg)(x)

Therefore, fg is an even function as it satisfies the definition of an even function, (fg)(x)=(fg)(x).

If f and g are odd functions, then by definition, f(x)=f(x) and g(x)=g(x)

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