Question

Determine whether the function is even, odd or neither. If is even or odd, use symmetry to sketch the graph.

$f\left(x\right)=x^4-4x^2$

Short answer

The function $f\left(x\right)$is even and the graph is:

Step-by-step solution

Step 1. Concept of even or odd function.

Letbe a function

f is even if$f(-x)=f\left(x\right)$for all x in the domain of f.

f is odd if$f(-x)=-f\left(x\right)$for all x in the domain of f.

The graph of an even function is symmetric with respect to y-axis.

The graph of an odd function is symmetric with respect to x-axis.

Step 2. Evaluate the function is even or odd.

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

Hence, f is an even function.

Step 3. Sketch the graph.

The graph of $f\left(x\right)$is symmetric about the y-axis.