Chapter 14.1, Problem 1MRE

To determine

To describe: The function is odd, even or neither,

  $f\left(x\right)=\frac{x+2}{x-3}$

Answer to Problem 1MRE

The given function is $neither$ odd nor even.

Explanation of Solution

Given information:

The function is odd, even or neither,

  $f\left(x\right)=\frac{x+2}{x-3}$

Concept used:

The Even function: The graph of $y=f\left(x\right)$ is systematic with respect to the $y-\text{axis}$ if, for every point $\left(x,y\right)$ on the graph, the point $\left(-x,y\right)$ is also on the graph that is, $f\left(-x\right)=f\left(x\right)$ for all $x$ in the domain of $f$ . Such a function is called even function.

Odd function: The graph of $y=f\left(x\right)$ is symmetric with respect to the origin if, for every point $\left(x,y\right)$ on the graph, the point $\left(-x,-y\right)$ is also on the graph that is, $f\left(-x\right)=f\left(x\right)$ for all $x$ in the domain of $f$ . Such a function is called an odd function.

Consider the function is odd, even or neither,

  $f\left(x\right)=\frac{x+2}{x-3}$

In the function replace $x=-x$ ,

Then the function is written as:

  $\begin{array}{c}f\left(x\right)=\frac{x+2}{x-3}\\ f\left(-x\right)=-\left(\frac{x-2}{x-3}\right)\\ f\left(-x\right)=\frac{-x-2}{-x+3}\end{array}$

Here,

  $\begin{array}{c}f\left(-x\right)\ne f\left(x\right)\\ f\left(-x\right)\ne -f\left(x\right)\end{array}$ .

So the function is neither even nor odd.

Thus, the given function is $neither$ odd nor even.