Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=\frac{x+9}{x-9} $$. Find the domain of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is all real $$ x $$, except $$ x= $$ B. The domain is all real $$ x $$. Find the $$ x $$-intercepts of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ x $$-intercept( $$ s $$ ) is/are at $$ x=\square $$. (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no $$ x $$-intercepts.

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To analyze the function $$ f(x) = \frac{x+9}{x-9} $$, we will follow these steps:

### Step 1: Find the Domain of $$ f(x) $$

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the function is undefined where the denominator is zero.

1. Set the denominator equal to zero:
   $$
   x - 9 = 0
   $$
   Solving this gives:
   $$
   x = 9
   $$

Thus, the function is undefined at $$ x = 9 $$. Therefore, the domain of $$ f(x) $$ is all real numbers except $$ x = 9 $$.

**Domain Choice:**
A. The domain is all real $$ x $$, except $$ x= 9 $$.

### Step 2: Find the $$ x $$-intercepts of $$ f(x) $$

The $$ x $$-intercepts occur where the function equals zero. To find the $$ x $$-intercepts, set the numerator equal to zero:

1. Set the numerator equal to zero:
   $$
   x + 9 = 0
   $$
   Solving this gives:
   $$
   x = -9
   $$

Thus, the $$ x $$-intercept is at $$ x = -9 $$.

**$$ x $$-intercept Choice:**
A. The $$ x $$-intercept(s) is/are at $$ x = -9 $$.

### Step 3: Sketch the Graph of $$ f(x) $$

To sketch the graph, we can identify some key features:

- **Vertical Asymptote:** The function has a vertical asymptote at $$ x = 9 $$ (where the function is undefined).
- **Horizontal Asymptote:** As $$ x $$ approaches infinity or negative infinity, the function approaches $$ y = 1 $$ (the leading coefficients of the numerator and denominator).
- **Intercepts:** The $$ x $$-intercept is at $$ (-9, 0) $$.

Now, let's summarize the information and sketch the graph.

### Summary of Information:
- **Domain:** All real $$ x $$, except $$ x = 9 $$.
- **$$ x $$-intercept:** $$ x = -9 $$.
- **Vertical Asymptote:** $$ x = 9 $$.
- **Horizontal Asymptote:** $$ y = 1 $$.

### Graph Sketch:
The graph will approach the vertical asymptote at $$ x = 9 $$ and will cross the $$ x $$-axis at $$ (-9, 0) $$. The function will approach the horizontal asymptote $$ y = 1 $$ as $$ x $$ goes to positive or negative infinity.

Now, I will proceed to sketch the graph of $$ f(x) $$.

### Graph of $$ f(x) $$:
- The graph will be in two parts: one part in the left of the vertical asymptote and one part in the right.
- The left part will approach the horizontal asymptote $$ y = 1 $$ as $$ x $$ goes to negative infinity and will cross the $$ x $$-axis at $$ (-9, 0) $$.
- The right part will approach the horizontal asymptote $$ y = 1 $$ as $$ x $$ goes to positive infinity and will go to positive infinity as it approaches the vertical asymptote at $$ x = 9 $$.

This completes the analysis of the function $$ f(x) = \frac{x+9}{x-9} $$.
**Summary:** - **Domain:** All real numbers except $$ x = 9 $$. - **$$ x $$-intercept:** $$ x = -9 $$. - **Graph:** Vertical asymptote at $$ x = 9 $$, horizontal asymptote at $$ y = 1 $$, and an $$ x $$-intercept at $$ (-9, 0) $$.

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