Chapter 4.5, Problem 110E

a.

To determine

To find whether h is even, odd or neither.

Answer to Problem 110E

  $h(x)={[f(x)]}^2=co{s}^{2}x$is even.

Explanation of Solution

Given information: It is given that $f(x)=\cos x$is an even function and $g(x)=\sin x$ is

an odd function and $h(x)={[f(x)]}^2$ .

Calculation:

  $h(x)={[f(x)]}^2=f(x).f(x)=\cos x.\cos x={\cos }^{2}x$

The graph of the $h(x)={\cos }^{2}x$ is given below.

  

It is clear from the graph $h(x)={\cos }^{2}x$ is even. Because all values of h(x) are positive and lies between 0 and 1.

So, $h(x)={[f(x)]}^2=co{s}^{2}x$is even.

b.

To determine

To find whether h is even, odd or neither.

Answer to Problem 110E

  $h(x)={[g(x)]}^2={\sin }^{2}x.$ is even.

Explanation of Solution

Given information: It is given that $f(x)=\cos x$is an even function and $g(x)=\sin x$ is

an odd function and $h(x)={[g(x)]}^2$ .

Calculation:

  $h(x)={[g(x)]}^2=g(x).g(x)=\sin x.\sin x={\sin }^{2}x.$

The graph of the $h(x)={\sin }^{2}x$ is given below.

  

It is clear from the graph $h(x)={\sin }^{2}x$ is even. Because all values of h(x) are positive and lies between 0 and 1.

So, $h(x)={[g(x)]}^2={\sin }^{2}x.$ is even.

c.

To determine

To find whether h is even, odd or neither.

Answer to Problem 110E

  $h(x)=f(x)g(x)=\cos x.\sin x$ is odd.

Explanation of Solution

Given information: It is given that $f(x)=\cos x$is an even function and $g(x)=\sin x$ is

an odd function and $h(x)=f(x)g(x)$ .

Calculation:

  $h(x)=f(x)g(x)=\cos x.\sin x$

  $$

.

The graph of the $h(x)=\sin x\cos x$ is given below.

  

It is clear from the graph $h(x)=\sin x\cos x$ is odd. Because all values of h(x) lies between 0.5 and -0.5.

So, $h(x)=f(x)g(x)=\cos x.\sin x$ is odd.