Problem 33 - 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. $$h(x)=\frac{x+7}{x^{2}+4 x-21}$$

Step-by-Step Solution

Answer

The function \(h(x)=\frac{x+7}{x^{2}+4 x-21}\) has a hole at \(x=-7\) and a vertical asymptote at \(x=3\).

Step 1: Factor the rational function

Factorize the polynomials in the numerator and the denominator of the rational function. The function \(h(x)=\frac{x+7}{x^{2}+4 x-21}\) can be factored into \(h(x)=\frac{x+7}{(x-3)(x+7)}\).

Step 2: Find the values of \(x\) for which the denominator becomes zero

Set the denominator equal to zero and solve for \(x\). This gives \(x-3=0\) and \(x+7=0\), thus the solution is \(x=3\) and \(x=-7\).

Step 3: Identify the asymptotes and the holes

The values of \(x\) for which the denominator is zero are \(x=3\) and \(x=-7\). However, since \(x=-7\) cancels out with the numerator, it results in a hole in the graph, not an asymptote. Thus, the graph has a hole at \(x=-7\) and a vertical asymptote at \(x=3\).