Q 44. - Precalculus Enhanced with Graphing Utilities

Question

In Problems 33– 44, determine algebraically whether each function is even, odd, or neither.

$F\left(x\right)=\frac{2x}{\left|x\right|}$

Step-by-Step Solution

Answer

The given function $F\left(x\right)=\frac{2x}{\left|x\right|}$ is an odd function.

Step 1. Write the given information and use the definition of odd and even function.

The given function is:

$F\left(x\right)=\frac{2x}{\left|x\right|}$

A function $f$is even if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f(-x)=f\left(x\right)$.

A function $f$ is odd if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f(-x)=-f\left(x\right)$.

Step 2. Determine if the function is even.

Replace $x$ by $-x$ in the given function,

$$\begin{gathered} F(-x)=\frac{2(-x)}{\left|-x\right|} \\ =\frac{-2x}{x} \end{gathered}$$

Since $F(-x)\ne F\left(x\right), F$ is not an even function.

Step 3. Determine if the function is odd.

Find the function $-F\left(x\right)$,

$-F\left(x\right)=\frac{-2x}{\left|x\right|}$

Since $F(-x)=-F\left(x\right), F$ is an odd function.