Problem 22 - College Algebra

Question

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

Step-by-Step Solution

Answer

The rational function \(f(x)=\frac{x}{x-3}\) has a vertical asymptote at \(x=3\) and no holes.

Step 1: Identify the denominator

The function being considered is \(f(x)=\frac{x}{x-3}\). The denominator of this function is \(x-3\).

Step 2: Find the values of \(x\) that make the denominator zero

To find the vertical asymptotes, set the denominator equal to zero and solve for \(x\). \[x-3=0\] Solving this equation, we find that \(x=3\).

Step 3: Determine if there are holes

A hole can appear in a rational function when a factor appears in both the numerator and denominator. But, in this case, the numerator is a constant and it doesn't have any factors that can cancel out with the denominator. So, there are no holes for this function.

Step 4: Identify the vertical asymptote

Since there are no removable discontinuities (holes), the zero in the denominator results in a vertical asymptote. Therefore, the vertical asymptote of this function is \(x=3\).