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homework assignment #6
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current learning objective: identifying vertical asymptotes of rational functions
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determine the vertical asymptotes and holes (removable points of discontinuity) of the rational function shown below.
f(x)=\frac{x - 6}{(x + 10)(x - 9)}
if there is not a hole or asymptote, record dne as your answer.
holes
x=
vertical asymptotes
(enter the asymptote with the smaller value first)
x=
x=

Explanation:

Step1: Find the domain restrictions

Set the denominator \((x + 10)(x - 9)=0\).
Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\).
So \(x+10 = 0\) gives \(x=-10\) and \(x - 9=0\) gives \(x = 9\).

Step2: Check for holes

A hole occurs when a factor in the numerator and a factor in the denominator cancel out.
The numerator is \(x - 6\). Since there is no common factor between \(x - 6\) and \((x + 10)(x - 9)\), there is no hole.

Step3: Determine vertical asymptotes

Vertical asymptotes occur at the values of \(x\) that make the denominator zero (when there is no common factor with the numerator).
Since \(x=-10\) and \(x = 9\) make the denominator zero and there are no common factors with the numerator, these are the vertical asymptotes.

Answer:

Holes: \(x=\text{DNE}\)
Vertical Asymptotes: \(x=-10\), \(x = 9\)