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Chapter 3: Problem 33

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x+7}{x^{2}+4 x-21}$$

Short Answer

Expert verified
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 by step solution

01

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 \(h(x)=\frac{x+7}{(x-3)(x+7)}\).
02

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 \(x=-7\).
03

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 \(x=3\).

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