Question

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

$f\left(x\right)=x^3-x$

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.

Let f be 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.

$$\begin{gathered} f\left(x\right)=x^3-x \\ f(-x)={\left(-x\right)}^3-\left(-x\right) \\ =-x^3+x \\ =-(x^3-x) \\ =-f\left(x\right) \end{gathered}$$

Hence, f is an odd function.

Step 3. Sketch the graph.

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