Problem 111 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.