Question

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

$f\left(x\right)=x+\left|x\right|$

Short answer

$f\left(x\right)$is neither even nor odd function.

Step-by-step solution

Given information.

The function,$f\left(x\right)=x+\left|x\right|$.

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$f\left(-x\right)=-f\left(x\right)$.

Calculate f-x.

Replace $x$by $-x$into the function $f\left(x\right)=x+\left|x\right|$.

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

Clearly, $f\left(-x\right)\ne f\left(x\right)andf\left(-x\right)\ne -f\left(x\right)$

So, by definition:$f\left(x\right)$is neither even nor odd function.