Chapter 2.2, Problem 102E

To determine

To identify: any symmetry of function with respect to x- axis, y -axis or origin and x- intercept of the function then draw graph of function.

Answer to Problem 102E

Function $f\left(x\right)=x^4-2x^2$ , has symmetry about $y$ axis.

Function $f\left(x\right)=x^4-2x^2$ has three $x$ intercepts.

Explanation of Solution

Given: Function is

  $f\left(x\right)=x^4-2x^2$

Calculation:

  $\begin{array}{l}f\left(x\right)=x^4-2x^2\\ =x^2\left(x^2-2\right)\end{array}$

i.e.

  $y=x^2\left(x^2-2\right)$

Function is symmetric about x axis if on replacing $y$ by $-y$ , equation of the function remain same.

On replacing, we get

  $-y=x^2\left(x^2-2\right)$

Hence graph of function is not symmetric with respect to x axis.

Now function is symmetric about y axis if on replacing $x$ by $-x$ , equation of the function remain same.

On replacing, we get

  $\begin{array}{l}y={\left(-x\right)}^2\left({\left(-x\right)}^2-2\right)\\ =x^2\left(x^2-2\right)\end{array}$

Hence graph of function issymmetric with respect to y axis.

Also function is symmetric about origin if on replacing $y$ by $-y$ and $x$ by $-x$ , equation of the function remain same.

On replacing, we get

  $\begin{array}{l}-y={\left(-x\right)}^2\left({\left(-x\right)}^2-2\right)\\ =x^2\left(x^2-2\right)\end{array}$

Hence graph of function is not symmetric with respect to origin.

For $x$ intercept, put

  $\begin{array}{l}f\left(x\right)=0\\ x^2\left(x^2-2\right)=0\\ \Rightarrow x=-\sqrt{2},0,\sqrt{2},\end{array}$

Hence $x$ intercept are $\left(-\sqrt{2},0\right)$ , $\left(0,0\right)$ and $\left(\sqrt{2},0\right)$ .

i.e. function has three $x$ intercepts.

Graph of function is