Chapter 1.5, Problem 71E

To determine

Whether the given function $f(x)=x^6-2x^2+3$ is even, odd or neither and then describe its symmetry.

Answer to Problem 71E

Given function $f(x)=x^6-2x^2+3$ is even and symmetric about $y-axis$ .

Explanation of Solution

Given information :

Given function $f(x)=x^6-2x^2+3$

As we have $f(x)=x^6-2x^2+3$

Replace $x$ by $-x$ in $f(x)$ ,

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

Thus, given function $f(x)=x^6-2x^2+3$ is even.

Since the function is even, so symmetry with respect to $y-axis$

To check symmetry about $x-axis$ , replace $y$ by $-y$ then $-y=-f(x)=x^6-2x^2+3$

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

This is not equivalent.

Thus, given function is not symmetry about $x-axis$ ,

Similar way, To check symmetry about origin, replace $x$ by $-x$ and $y$ by $-y$ then

  $\begin{array}{l}\Rightarrow -y=({\left(-x\right)}^6-2{\left(-x\right)}^2+3)\\ \Rightarrow -y=f(x)\\ \therefore y=-f(x)\end{array}$

This is also not equivalent.

Thus, given function is not symmetry about origin,

Hence, given function $f(x)=x^6-2x^2+3$ is even and symmetric about $y-axis$ .