Q.37 - Precalculus Enhanced with Graphing Utilities

Question

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

$F\left(x\right)=\sqrt[3]{x}$

Step-by-Step Solution

Answer

$F\left(x\right)$ is an odd function.

Step 1: Given information.

The function, $F\left(x\right)=\sqrt[3]{x}$

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 $F\left(x\right)=\sqrt[3]{x}$.

$$\begin{gathered} F\left(-x\right)=\sqrt[3]{-x} \\ =-\sqrt[3]{x} \\ =-F\left(x\right) \end{gathered}$$

Clearly, $F\left(-x\right)=-F\left(x\right)$

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