Chapter 1.5, Problem 99E

a .

To determine

To find:whether $g\left(x\right)=-f\left(x\right)$ is even, odd, or neither.

Answer to Problem 99E

Even

Explanation of Solution

Given:

The function f is an even function.

Concept Used:

A function $f\left(x\right)$ is said to be an even function if for all x in the domain of f , $f\left(-x\right)=f\left(x\right)$ .

A function $f\left(x\right)$ is said to be an odd function if for all x in the domain of f , $f\left(-x\right)=-f\left(x\right)$ .

It is known that $f\left(x\right)$ is an even function.

So, $f\left(-x\right)=f\left(x\right)$

Now substitute $-x$ for x in $g\left(x\right)$ .

  $\begin{array}{c}g\left(-x\right)=-f\left(-x\right)\\ =-f\left(x\right)\\ =g\left(x\right)\end{array}$

Thus, $g\left(x\right)$ is an even function.

b.

To determine

To find: whether $g\left(x\right)=f\left(-x\right)$ is even, odd, or neither.

Answer to Problem 99E

Even

Explanation of Solution

Given:

The function f is an even function.

Concept Used:

A function $f\left(x\right)$ is said to be an even function if for all x in the domain of f , $f\left(-x\right)=f\left(x\right)$ .

A function $f\left(x\right)$ is said to be an odd function if for all x in the domain of f , $f\left(-x\right)=-f\left(x\right)$ .

It is known that $f\left(x\right)$ is an even function.

So, $f\left(-x\right)=f\left(x\right)$

Consider,

  $g\left(x\right)=f\left(-x\right)$

But $f\left(-x\right)=f\left(x\right)$ , this implies that

  $g\left(x\right)=f\left(x\right)$ ….. (1)

Now To find if g is even or odd, substitute $-x$ for x in $g\left(x\right)$ .

  $\begin{array}{c}g\left(-x\right)=f\left(-\left(-x\right)\right)\\ =f\left(x\right)\end{array}$

But from (1), $g\left(x\right)=f\left(x\right)$

This implies that,

  $g\left(-x\right)=g\left(x\right)$

Thus, $g\left(x\right)$ is an even function.

c.

To determine

To find: whether $g\left(x\right)=f\left(x\right)-2$ is even, odd, or neither.

Answer to Problem 99E

Even

Explanation of Solution

Given:

The function f is an even function.

Concept Used:

A function $f\left(x\right)$ is said to be an even function if for all x in the domain of f , $f\left(-x\right)=f\left(x\right)$ .

A function $f\left(x\right)$ is said to be an odd function if for all x in the domain of f , $f\left(-x\right)=-f\left(x\right)$ .

It is known that $f\left(x\right)$ is an even function.

So, $f\left(-x\right)=f\left(x\right)$

Now To find if g is even or odd, substitute $-x$ for x in $g\left(x\right)$ .

  $\begin{array}{c}g\left(-x\right)=f\left(-x\right)-2\\ =f\left(x\right)-2\\ =g\left(x\right)\end{array}$

Thus, $g\left(x\right)$ is an even function.

d.

To determine

To find: whether $g\left(x\right)=f\left(x-2\right)$ is even, odd, or neither.

Answer to Problem 99E

Neither

Explanation of Solution

Given:

The function f is an even function.

Concept Used:

If a function is even it is symmetric with respect to the y -axis and if it is odd then it is symmetric with respect to the origin.

It is known that $f\left(x\right)$ is an even function. So it is symmetric with respect to the y -axis.

Now graph of $g\left(x\right)=f\left(x-2\right)$ is the translation of the graph of $f\left(x\right)$ two units to the right.

And the graph thus obtained is neither symmetric to y -axis nor to the origin.

Thus, $g\left(x\right)$ is neither even nor odd.