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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 there is one, of the function's graph.

Short Answer

Expert verified
Vertical asymptotes of a rational function can be found in places where the denominator equals zero while the numerator does not. To find these places, set the denominator equal to zero and solve it for x, then insert these x-values into the numerator. If the numerator does not equal to zero for any of these x-values, then they are the function's vertical asymptotes.

Step by step solution

01

Understand the Vertical Asymptote

A vertical asymptote of a rational function is a vertical line x = a if the function heads towards 卤 infinity as x approaches a from the left or right. This usually happens when dividing by zero in a function's equation.
02

Identify the Rational Function

The rational function is represented as \(f(x) = P(x) / Q(x)\), where \(P(x)\) and \(Q(x)\) are polynomials. The function \(f(x)\) has a vertical asymptote at \(x = a\) if \(Q(a) = 0\) and \(P(a) 鈮� 0\).
03

Solve for Zero

To actually find if a function has a vertical asymptote, first, set the denominator equal to zero and solve for x. These will be the possible vertical asymptotes.
04

Check the Numerator

Insert the obtained x-values back into the numerator of the function. If the numerator is not equal to zero for any of the x-values, then these are the vertical asymptotes of the function. Any x-value that results in a zero in the numerator is a hole in the graph, not a vertical asymptote.

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