Question

Determine the values for which the rational expression is undefined.

$\frac{\mathrm{x}-3}{{\mathrm{x}}^2-\mathrm{x}-30}$

Short answer

Rational function is undefined when$\mathrm{x}=6,-5.$

Step-by-step solution

Step 1. Given Data

Given that the rational function is$\mathrm{R}\left(\mathrm{x}\right)=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-\mathrm{x}-30}.$

Step 2. Solving quadratic equation

On factorizing the denominator using AC method,

$$\begin{gathered} {\mathrm{x}}^2-\mathrm{x}-30 \\ {\mathrm{x}}^2-6\mathrm{x}+5\mathrm{x}-30 \\ \mathrm{x}(\mathrm{x}-6)+5(\mathrm{x}-6) \\ (\mathrm{x}-6)(\mathrm{x}+5) \\ \mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-3)}{(\mathrm{x}-6)(\mathrm{x}+5)} \end{gathered}$$

Step 3. Critical point

To find the critical point,

$$\begin{gathered} (\mathrm{x}-3)=0 \\ \mathrm{x}=3 \\ (\mathrm{x}-6)(\mathrm{x}+5)=0 \\ \mathrm{x}=6,-5 \end{gathered}$$

The denominator can be substituted using critical points when the function is undefined.

$$\begin{gathered} \mathrm{At}\mathrm{x}=6,\mathrm{R}\left(\mathrm{x}\right)=\frac{6-3}{0}=\infty \\ \mathrm{At}\mathrm{x}=-5,\mathrm{R}\left(\mathrm{x}\right)=\frac{-5-3}{0}=\infty \end{gathered}$$

The rational function can be undefined at x=6,-5.