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Chapter 3: Problem 111

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

Short Answer

Expert verified
To find the vertical asymptotes of a rational function, set the denominator of the function equal to zero and solve for x. The solution(s) is the potential vertical asymptote(s). Then plug these x-values into the numerator of the function. If the numerator of the function also becomes zero at any of these x-values, then this point is not a vertical asymptote, otherwise, it is.

Step by step solution

01

Identify the rational function

Firstly, analyse and understand the given rational function. A rational function is of the form \( \frac {f(x)}{g(x)} \) where \( f(x) \) and \( g(x) \) are polynomial functions.
02

Set the denominator equal to zero

To find the vertical asymptotes, set the denominator \( g(x) \) equal to zero. That is, solve the equation \( g(x) = 0 \). These x-values are the potential vertical asymptotes.
03

Check the numerator at these x-values

Then, for each x-value found in Step 2, plug it into the numerator of the function \( f(x) \). If \( f(x) \) is also zero at this x-value, then it indicates that this is not a vertical asymptote, but might be a removable discontinuity.
04

Confirming the vertical asymptotes

If at any x-value found in Step 2, the value of \( f(x) \) is not 0, then it confirms that this x-value is a vertical asymptote of the function.

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