Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

Determine(algebraically)whethere the given function is even, odd,or neither.

$g\left(x\right)=\frac{4+x^2}{1+x^4}$

Step-by-Step Solution

Answer

The given $g\left(x\right)=\frac{4+x^2}{1+x^4}$is an even function

Step 1.Given information

The given function $g\left(x\right)=\frac{4+x^2}{1+x^4}$

Step 2.Determine the given function is even,odd or neither.

The function is even if $g(-x)=g\left(x\right)$ and odd if $g(-x)=-g\left(x\right)$.If none of these conditions are satisfied.

Find $g(-x)$ by replacing -x for x in the given function.

$$\begin{gathered} g(-x)=\frac{4+{(-x)}^2}{1+{(-x)}^4} \\ =\frac{4+x^2}{1+x^2} \\ =g\left(x\right) \end{gathered}$$

Since $g(-x)=g\left(x\right)$.We can conclude that the function $g\left(x\right)$ is even.