Q 41. - Precalculus Enhanced with Graphing Utilities

Question

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

$g\left(x\right)=\frac{x^2+3}{x^2-1}$

Step-by-Step Solution

Answer

The given function $g\left(x\right)=\frac{x^2+3}{x^2-1}$ is even.

Step 1. Write the given information and use the definition of odd and even function.

The given function is:

$g\left(x\right)=\frac{x^2+3}{x^2-1}$

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 function:

$g\left(x\right)=\frac{x^2+3}{x^2-1}$

$$\begin{gathered} g(-x)=\frac{{(-x)}^2+3}{{(-x)}^2-1} \\ =\frac{x^2+3}{x^2-1} \end{gathered}$$

Since $h(-x)=h\left(x\right)$, the function is even.