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Chapter 3: Problem 24

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x-3)}$$

Short Answer

Expert verified
The vertical asymptotes of the function \(g(x)=\frac{x+3}{x(x-3)}\) are at \(x=0\) and \(x=3\). There are no holes in this function.

Step by step solution

01

Identify Potential Asymptotes and Holes

Potential vertical asymptotes or holes will occur where the denominator \(x(x-3)\) is equal to zero. This is when \(x=0\) or \(x=3\).
02

Test for Cancellation

A hole occurs when the same non-zero divisor factor can be found in the denominator and numerator. In this function, the numerator does not have the same factors that could cancel out any factors from the denominator. Therefore, there are no holes.
03

Determine Vertical Asymptotes

Since there are no cancellations, the values \(x=0\) and \(x=3\) are vertical asymptotes of the function. This is due to division by zero at these points.

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