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Chapter 3: Problem 34

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x+6}{x^{2}+2 x-24}$$

Short Answer

Expert verified
For \(h(x) = \frac{x+6}{x^{2}+2x-24}\), \(x=4\) is a vertical asymptote and \(x=-6\) corresponds to a hole in the graph.

Step by step solution

01

Identify potential vertical asymptotes and holes

Vertical asymptotes and holes occur where the denominator of the rational function is equal to zero. We begin by setting the denominator of \(h(x)\) equal to zero and solving the resulting quadratic equation.\(x^{2}+2x-24=0\)
02

Finding the zeroes of the denominator

The quadratic equation from step 1 can be factored and then solved to find the zeroes.\(x^{2}+2x-24=(x-4)(x+6)\)Setting these factors to 0:\(x-4=0\) gives \(x=4\)\(x+6=0\) gives \(x=-6\)
03

Identifying if these are vertical asymptotes or holes

To be a vertical asymptote, \(x=4\) or \(x=-6\) has to make the numerator (\(x+6\)) equal to a non-zero number. For \(x=4\), \(x+6 = 4+6 = 10\), which is not zero. To be a hole, \(x=4\) or \(x=-6\) has to make both the numerator and the denominator equal to zero. We have seen that for \(x=-6\), both the numerator (\(x+6\)) and the denominator (\(x^{2}+2x-24\)) equal zero.

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)-\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x-3,\) a horizontal asymptote \(y-0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

Among all pairs of numbers whose difference is 24 . find a pair whose product is as small as possible. What is the minimum product?

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{3}{x+3}>\frac{3}{x-2} $$

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the function's graph.

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\)
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