To analyze the rational function [tex]\( f(x) = \frac{x}{x-3} \)[/tex] for vertical asymptotes and holes, let's go through the steps in detail:
### Step 1: Find the Vertical Asymptotes
Vertical asymptotes occur where the denominator of the rational function equals zero, causing the function to be undefined.
Given [tex]\( f(x) = \frac{x}{x-3} \)[/tex], we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 3 \)[/tex].
### Step 2: Find the Holes
Holes in the graph of a rational function occur where there are common factors in the numerator and the denominator that cancel each other out.
For the function [tex]\( f(x) = \frac{x}{x-3} \)[/tex], the numerator is [tex]\( x \)[/tex] and the denominator is [tex]\( x-3 \)[/tex]. Clearly, there are no common factors between the numerator and the denominator. Therefore, there are no holes in the graph of this function.
### Conclusion
Based on this analysis, the correct choice is:
C. The vertical asymptote(s) is (are) [tex]\( x = 3 \)[/tex]. There are no holes.
Thus, the final answer is:
C. The vertical asymptote(s) is (are) [tex]\( x = 3 \)[/tex]. There are no holes.
