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

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1.1, Problem 74E
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.

Expert Solution

The function fg is an even function if f and g are even.

The function fg is an even function if f and g are odd.

The function fg is an odd function if f is even and g is odd.

### Explanation of Solution

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).

As f and g are odd 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))

Given that f is an even function and g is an odd function.

Then by the definition, f(x)=f(x) and g(x)=g(x).

Substitute f(x)=f(x) and g(x)=g(x) in (1),

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

Observe that the function fg(x)=(fg)(x) satisfies the definition of an odd function.

Therefore, the function fg is an odd function when f is an even function and g is an odd function

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