Q.36 - Precalculus Enhanced with Graphing Utilities

Question

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

$h\left(x\right)=3x^3+5$

Step-by-Step Solution

Answer

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

Step 1: Given information.

The function: $h\left(x\right)=3x^3+5$.

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 h - x .

Replace $x$ by $-x$ into the function $h\left(x\right)=3x^3+5$.

$$\begin{gathered} h\left(-x\right)=3{\left(-x\right)}^3+5 \\ =-3x^3+5 \end{gathered}$$

is neither even nor odd.

Clearly,  $h\left(-x\right)\ne h\left(x\right) and h\left(-x\right)\ne -h\left(x\right)$

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