Q. 33 - Precalculus Enhanced with Graphing Utilities

Question

Determine algebraically whether each function is even, odd, or neither.

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

Step-by-Step Solution

Answer

$f\left(x\right)$ is an odd function.

Step 1: Given information.

The function $f\left(x\right)=4x^3$.

Step 2: Write definition of even function and odd function.

• A function $f\left(x\right)$ is said to be an even function if $f\left(-x\right)=f\left(x\right)$.
• A function $f\left(x\right)$ is said to be an odd function if $f\left(-x\right)=-f\left(x\right)$.

Step 3: Calculate f - x .

Replace $x$ by $-x$ into the given function $f\left(x\right)=4x^3$.

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

As, $f\left(-x\right)=-f\left(x\right)$

So, by definition: $f\left(x\right)$ is an odd function.