Chapter 1.5, Problem 97E

a.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The function is odd as function is symmetric with respect to origin.

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x$

  

A function $f$ is odd only if $f\left(-x\right)=-f\left(x\right)$

Hence, the function is odd as function is symmetric with respect to origin.

b.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The function is even as function is symmetric with respect to $y-axis$ .

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x^2$

  

A function $f$ is odd only if $f\left(-x\right)=f\left(x\right)$ .

Hence, the function is even as function is symmetric with respect to $y-axis$ .

c.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The function is odd as function is symmetric with respect to origin

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x^3$

  

A function $f$ is odd only if $f\left(-x\right)=-f\left(x\right)$ .

Hence, the function is odd as function is symmetric with respect to origin.

d.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The function is even as function is symmetric with respect to $y-axis$ .

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x^4$

  

A function $f$ is odd only if $f\left(-x\right)=f\left(x\right)$ .

Hence, the function is even as function is symmetric with respect to $y-axis$ .

e.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The function is odd as function is symmetric with respect to origin.

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x^5$

  

A function $f$ is odd only if $f\left(-x\right)=-f\left(x\right)$

The function is odd as function is symmetric with respect to origin.

f.

To determine

Describe similarity and difference among the graphs.

Answer to Problem 97E

The graphs of even functions are symmetric with respect to $y-axis$ .

Explanation of Solution

Calculation:

Let us plot the graph of following function:

  $y=x^6$

  

A function $f$ is odd only if $f\left(-x\right)=f\left(x\right)$ .

The function is odd as function is symmetric with respect to $y-axis$ .

The function with even power of is even functions and with odd power of is odd function.

Hence, The functions $y=x^2$ , $y=x^4$ and $y=x^6$ are even functions. The graphs of even functions are symmetric with respect to $y-axis$ . That is, after rotation about the $y-axis$ , the graph remains unchanged. The functions $y=x$ , $y=x^3$ and $y=x^5$ are odd functions. The graphs of odd functions are symmetric with respect to origin . That is, after rotation of $180$ degrees about the origin the graph remains unchanged.