To Check : whether h is necessarily even function if h = f ∘ g and g is an even function. To check if h is odd if g is odd. To check if h is even or odd if g is odd and f is odd. To check if h is even or odd if g is odd and f is even.

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 2.6, Problem 70E
To determine

To Check : whether h is necessarily even function if h=f∘g and g is an even function. To check if h is odd if g is odd. To check if h is even or odd if g is odd and f is odd. To check if h is even or odd if g is odd and f is even.

Expert Solution

Function h is necessarily even, if g is even. If g is odd, the nature of h depends on f. If both g and f are odd, h is odd. If g is odd and f is even, h is even.

Explanation of Solution

Given information : Given h=fg . In part 1, g is an even function. In part 2, g is odd function. In part 3, g is odd and f is odd. In part 4, g is odd and f is even.

Formula used : The concept of Composition of Functions and Odd/ Even Functions is used.

Calculation :

In part 1, g is even function

g(x)=g(x)

So, for h to be even function h(x)=h(x)

h(x)=f(g(x))

And g(x)=g(x) , so h(x)=f(g(x))=h(x)

Thus, h is necessarily even function, if g is even function.

In part 2, g is odd function

g(x)=g(x)

So, for h to be even function h(x)=h(x)

h(x)=f(g(x))h(x)=f(g(x))

Thus, as the nature of f is not known, we can’t say whether h is even or odd.

In part 3, g is odd function and f is odd

g(x)=g(x) and f(x)=f(x)

So, for h to be even function h(x)=h(x)

h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x))=h(x)

Thus, h is an odd function, if g is odd function and f is odd function.

In part 4, g is odd function and f is even

g(x)=g(x) and f(x)=f(x)

So, for h to be even function h(x)=h(x)

h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x))=h(x)

Thus, h is a even function, if g is odd function and f is even function.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!