Question

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

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

Short answer

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

Step-by-step solution

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 role="math" localid="1645811873270" $F(-x)\ne F\left(x\right),F$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.