Chapter 1.3, Problem 109E

1.

To determine

To find : the function g(x).

Answer to Problem 109E

Thus the function is even.

Explanation of Solution

Given information: f is an even function. And g(x)=-f(x)

Calculation: given that, f(x) is even function

we assume , $f(x)=x^n$ , n is even number .

then $g(x)=-f(x)$

  $\begin{array}{l}=f(-x)\\ $ ={(-x)}^n$=x^n\end{array}$

  $\therefore g(x)$ is an even function.

2.

To determine

To find : the function g(x).

Answer to Problem 109E

Thus g(x) is even.

Explanation of Solution

Given information:

f(x) is even

g(x)=f(-x)

Calculation: given that ,$f(x)$is even number$f(x)=x^n$

  $\therefore f(-x)={(-x)}^n=x^n$

  $\therefore g(x)=f(-x)=x^n$

Thus g(x) is even.

3.

To determine

To find : the function g(x)

Answer to Problem 109E

Thus g(x) is even.

Explanation of Solution

Given information:

f(x) is even

  $g(x)=f(x)-2$

Calculation: given that,

  $f(x)$ be an even function.

Let , $f(x)=x^n+2$ , n is even number.

Then , $g(x)=f(x)-2$

  $\therefore g(x)=x^n+2-2$

  $=x^n$ , n is even number,

This, g(x) is even function.

4.

To determine

To find : the function g(x)

Answer to Problem 109E

Thus g(x) is odd.

Explanation of Solution

Given information:

f(x) is even function.

g(x)=f(-x)

Calculation:

given that , f(x) is even .

let, $f(x)=x^n$ , n is even number.

  $\therefore g(x)=-f(x+3)$

  $={\{-(x+3)\}}^n$

  $={(x+3)}^n$

Thus, g(x) is odd function.