Problem 32 - College Algebra

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x-5}{x^{2}-25}$$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{x-5}{x^{2}-25}\) has a vertical asymptote at \(x=-5\) and a hole at \(x=5\).

Step 1: Factor the Function

Firstly, factor the denominator as well as the numerator. The given function \(g(x)=\frac{x-5}{x^{2}-25}\) can be factored as \(g(x)=\frac{x-5}{(x-5)(x+5)}\). By comparing the numerator and the denominator, it can be observed that \(x-5\) is common in both the numerator and the denominator.

Step 2: Find the Vertical Asymptotes

To find the vertical asymptotes, equate the denominator to zero and solve for \(x\). The denominator is \((x-5)(x+5)\). Set it equal to zero, that gives: \((x-5)(x+5)=0\). Solving for \(x\), we get \(x=5\) and \(x=-5\). However, since \(x-5\) is in both the numerator and denominator, \(x=5\) will be a hole instead of a vertical asymptote. Hence, there is one vertical asymptote at \(x=-5\).

Step 3: Find the Holes

A hole exists when a value makes both the numerator and denominator of the function zero. As observed in Step 1, \(x-5\) is common to both the numerator and the denominator. Thus, \(x=5\) is a hole in the function.