ummarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=\frac{x+3}{x-3} $$ ind any horizontal asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, $$ \square $$ (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is $$ \square $$ (Type equations.) . C. There are no horizontal asymptotes. nd any vertical asymptotes of $$ f(x) $$ Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, $$ \square $$ (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $$ \square $$ and the rightmost asymptote is $$ \square $$ . (Type equations.) C. There are no vertical asymptotes.

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To analyze the function $$ f(x) = \frac{x+3}{x-3} $$, we will determine its horizontal and vertical asymptotes and sketch the graph.

### Step 1: Identify Horizontal Asymptotes

Horizontal asymptotes are determined by the behavior of the function as $$ x $$ approaches infinity or negative infinity.

For rational functions of the form $$ \frac{a_n x^n + \ldots}{b_m x^m + \ldots} $$:
- If $$ n < m $$, the horizontal asymptote is $$ y = 0 $$.
- If $$ n = m $$, the horizontal asymptote is $$ y = \frac{a_n}{b_m} $$.
- If $$ n > m $$, there is no horizontal asymptote.

In our case:
- The degree of the numerator $$ n = 1 $$ (from $$ x+3 $$).
- The degree of the denominator $$ m = 1 $$ (from $$ x-3 $$).

Since $$ n = m $$, we find the horizontal asymptote as follows:
$$
y = \frac{1}{1} = 1
$$

### Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).

Set the denominator equal to zero:
$$
x - 3 = 0 \implies x = 3
$$

### Summary of Asymptotes

- **Horizontal Asymptote**: $$ y = 1 $$
- **Vertical Asymptote**: $$ x = 3 $$

### Step 3: Sketch the Graph

1. The function approaches the horizontal asymptote $$ y = 1 $$ as $$ x $$ approaches $$ \pm \infty $$.
2. The function has a vertical asymptote at $$ x = 3 $$, where the function is undefined.
3. The function will cross the horizontal asymptote at some points, which can be found by setting $$ f(x) = 1 $$:
   $$
   \frac{x+3}{x-3} = 1 \implies x + 3 = x - 3 \implies 3 = -3 \quad \text{(no solutions)}
   $$
   Thus, it does not cross the horizontal asymptote.

### Final Answers

- **Horizontal Asymptote**: 
  - A. The function has one horizontal asymptote, $$ y = 1 $$.

- **Vertical Asymptote**: 
  - A. The function has one vertical asymptote, $$ x = 3 $$.

### Conclusion

The pertinent information is summarized as follows:
- The function has one horizontal asymptote at $$ y = 1 $$.
- The function has one vertical asymptote at $$ x = 3 $$.
The function $$ f(x) = \frac{x+3}{x-3} $$ has one horizontal asymptote at $$ y = 1 $$ and one vertical asymptote at $$ x = 3 $$.

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