Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

Determine(algebraically)whethere the given function is even, odd,or neither.

$f\left(x\right)=x^3-4x$

Step-by-Step Solution

Answer

The given function $f\left(x\right)=x^3-4x$ is odd function

Step 1.Given information

The given function $f\left(x\right)=x^3-4x$

Step 2.Determine whether the given function is even, odd,or neither.

The function is even  if $f(-x)=f\left(x\right)$ and odd  if $f(-x)=-f\left(x\right)$ .If none of these conditions are satisfied.

Find $f(-x)$ by replacing -x for x in the given function

$$\begin{gathered} f(-x)={(-x)}^3-4(-x) \\ =-x^3+4x \\ =-(x^3-4x) \\ =-f\left(x\right) \end{gathered}$$

Therefore,the given function is odd. since,$f(-x)=-f\left(x\right)$