Q. 35 - Precalculus Enhanced with Graphing Utilities

Question

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

$g\left(x\right)=-3x^2-5$

Step-by-Step Solution

Answer

$g\left(x\right)$ is an even function.

Step 1: Given information.

The function: $g\left(x\right)=-3x^2-5$

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 g - x .

Replace $x$ by $-x$ into the given function $g\left(x\right)=-3x^2-5$.

$$\begin{gathered} g\left(-x\right)=-3{\left(-x\right)}^2-5 \\ =-3x^2-5 \\ =g\left(x\right) \end{gathered}$$

As, $g\left(-x\right)=g\left(x\right)$

So by definition, $g\left(x\right)$ is an even function.