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Find the vertical asymptotes, if any, and the values of [tex]x[/tex] corresponding to holes, if any, of the graph of the rational function.

[tex]\[ f(x) = \frac{x}{x-3} \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type an equation. Use commas to separate answers as needed.)

A. The vertical asymptote(s) is (are) [tex] \square [/tex] and hole(s) corresponding to [tex] \square [/tex].

B. There are no vertical asymptotes but there is (are) hole(s) corresponding to [tex] \square [/tex].

C. The vertical asymptote(s) is (are) [tex] \square [/tex]. There are no holes.

D. There are no discontinuities.



Answer :

GinnyAnswer
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.

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