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Chapter 2: Problem 115

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
The vertical asymptotes of a rational function can be found by setting the denominator equal to zero and solving for \( x \) in the equation. Subsequently, verify that the numerator isn鈥檛 zero for the same \( x \)-values. The valid values obtained are the vertical asymptotes of the function.

Step by step solution

01

Identify the Function

To form the rational function, let's assume the generic form of the function is \( f(x) = \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials.
02

Set the Denominator Equals to Zero

In order to find the vertical asymptotes, set the denominator of the function equals to zero. This can be written as \( Q(x) = 0 \)
03

Solve for x in the Equation

Solve the equation \( Q(x) = 0 \) for \( x \) to find the values of \( x \) which satisfy the equation. These \( x \) values will be potential vertical asymptotes for the function.
04

Check the Numerator

To confirm the vertical asymptotes, substitute each \( x \) value obtained in step 3 into \( P(x) \). If the result for \( P(x) \) is non-zero at this \( x \)-value, then \( x \) is a vertical asymptote. If it does equal zero, then \( x \) is not a vertical asymptote.

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