Problem 24
Question
Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x-3)}$$
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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 \(x=3\).
2Step 2: 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.
3Step 3: 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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