Question

Determine whether the function is even, odd, or neither.

$f\left(x\right)=x{\sin }^{3}x$

Short answer

The$f\left(x\right)=x{\sin }^{3}x$is an even function.

Step-by-step solution

Step 1. State the definitions of even and odd functions.

A function $f\left(x\right)$is an even function if $f\left(x\right)=f\left(-x\right)$.

A function $f\left(x\right)$ is an odd function if $f\left(x\right)=f\left(-x\right)$.

Step 2. Calculate f−x. 

Substitute $-x$for $x$into the function $f\left(x\right)=x{\sin }^{3}x$and simplify.

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

Step 3. Simplify.

Note: $\sin \left(-t\right)=-\sin t$, that is., sine function, in general, is an odd function.

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

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

Since, $f\left(-x\right)=f\left(x\right)$, therefore, $f\left(x\right)=x{\sin }^{3}x$ is an even function.