# To check: Whether h = f ∘ g is always an odd function if g is odd; also check what would be the result if f is an odd function and if f is even function. ### 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 1.3, Problem 64E
To determine

## To check: Whether h=f∘g is always an odd function if g is odd; also check what would be the result if f is an odd function and if f is even function.

Expert Solution

The composition function h=fg is not always an odd function.

Iff is an odd function, then the composition function h=fg is always an odd function.

If f is an even function, then the composition function h=fg is always an even function.

### Explanation of Solution

Given:

The function g(x) is an odd function.

Therefore, g(x)=g(x) .

The composite function is, h(x)=(fg)(x)

That is h(x)=f(g(x)) .

Verification:

The composition function h(x) is said to be an odd function if h(x)=h(x) , is said to be an even function if h(x)=h(x) .

So, first obtain the value of h(x) as shown below.

h(x)=f(g(x))=f(g(x))              [g(x)=g(x)]

Thus, h(x)=f(g(x)) (1)

Here, it cannot be concluded that whether the composition function h(x) is an even function or an odd function as it depends on the nature of the function f(x) . Thus, h(x) is not always an odd function.

There exist two cases which are explained as follows.

Case 1:

If f(x) is an odd function, then the equation (1) becomes,

h(x)=f(g(x))=f(g(x))              [f(x)=f(x)]=h(x)

Thus, the composite function h(x)=(fg)(x) is an odd function when the functions f(x) and g(x) are odd functions.

Case 2:

If f(x) is an even function, then the equation (1) becomes,

h(x)=f(g(x))=f(g(x))              [f(x)=f(x)]=h(x)

Thus, the composite function h(x)=(fg)(x) is an even function when the function f(x) is even and the function g(x) is odd functions.

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