Chapter 7, Problem 1MCP

To determine

All the numbers for which the given rational function is undefined.

Answer to Problem 1MCP

Solution: The numbers that make the given function undefined are: $x=-2,4$ .

Explanation of Solution

Given:

The rational function $f\left(x\right)=\frac{x^2-4}{x^2-2x-8}$ .

Key concepts used:

A rational function is undefined when its denominator becomes zero.

Calculation:

Since we know that a rational function is only defined if its denominator is nonzero. Therefore, in order to find the values of x for which the given rational function is undefined, we will set the denominator of the function equal to zero and then we will solve for x.

  $\text{The function }f\left(x\right)=\frac{x^2-4}{x^2-2x-8}\text{ has denominator }{x}^2-2x-8.$

Upon setting the denominator of the given rational function equal to zero, we get:

  $\begin{array}{l}\Rightarrow x^2-2x-8=0\\ \Rightarrow x^2-4x+2x-8=0\left(\begin{array}{l}\text{Upon splitting the }\\ \text{middle term}\end{array}\right)\\ \Rightarrow x\left(x-4\right)+2\left(x-4\right)=0\left(\begin{array}{l}\text{Upon pulling out GCF }\\ \text{from each parenthesis}\end{array}\right)\\ \Rightarrow \left(x-4\right)\left(x+2\right)=0\left(\begin{array}{l}\text{Upon pulling out the}\\ \text{common factor}\end{array}\right)\\ \Rightarrow \left(x-4\right)=0\text{ or }\Rightarrow \left(x+2\right)=0\left(\begin{array}{l}\text{Upon using zero }\\ \text{product property}\end{array}\right)\\ \Rightarrow x=-2,4\end{array}$

Conclusion:

We used the fact that a rational function is undefined when its denominator is equal to zero and found the required values as $x=-2,4$ .