Chapter 3.7, Problem 3CFU

To determine

To find: an equation for the given situation.

Answer to Problem 3CFU

  $f\left(x\right)=\frac{x-2}{x^2-4}$

Explanation of Solution

Given information:

The given situation is ‘a rational function that has point of discontinuity’.

Calculation:

The function $f\left(x\right)$ given below has zeros at $x=\pm 2$ .

  $f\left(x\right)=\frac{x-2}{x^2-4}$

The factor $\left(x-2\right)$ is removable. Thus, the function is simplified as follows:

  $\begin{array}{c}f\left(x\right)=\frac{x-2}{x^2-4}\\ =\frac{x-2}{\left(x-2\right)\left(x+2\right)}\left[\text{Simplifying}\right]\\ =\frac{1}{x+2}\end{array}$

The graph for $f\left(x\right)$ is shown below.

  

Since, the original function had a discontinuity at $x=2$ it is still recognized with a hole at that point. Thus, the rational function with a point of discontinuity is $f\left(x\right)=\frac{x-2}{x^2-4}$ .