Chapter 1.5, Problem 71E

To determine

To calculate; function is even, odd or neither and then describe the symmetry.

Answer to Problem 71E

Solution: Even function, symmetric about $y-$ axis.

Explanation of Solution

  $f(x)=x^6-2x^2+3$

Let $y=x^6-2x^2+3$

Compute $f(-x)$ , because if $f(x)=f(-x)$ , then the function is even.

  $\begin{array}{l}$ f(-x)={(-x)}^6-2{(-x)}^2+3$f(-x)=x^6-2x^2+3=f(x)\end{array}$

Hence, the function is even.

For the symmetric check about $x-$ axis replace $y$ by $-y.$

  $\begin{array}{l}-y=x^6-2x^2+3\\ y=-x^6+2x^2-3\end{array}$

The result is not equivalent equation hence it is not symmetric about x-axis.

For the symmetric check about $y-$ axis replace $x$ by $-x.$

  $\begin{array}{l}$ y={(-x)}^6-2{(-x)}^2+3$y=x^6-2x^2+3\end{array}$

The result is equivalent equation hence there is symmetry about the $y-$ axis.

The symmetry about the origin is checked by replacing $x$ by $-x.$ and $y$ by $-y.$

  $\begin{array}{l}$ y={(-x)}^6-2{(-x)}^2+3$y=-x^6+2x^2-3\end{array}$

The result is not equivalent equation hence no symmetry about origin.