Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of the function given below. $$ f(x)=\frac{9 x}{1-x^{2}} $$ Summarize the pertinent information obtained by analyzing $$ f(x) $$. 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 of $$ f(x) $$ is all real $$ x $$, except $$ x=\square $$. (Use a comma to separate answers as needed.) B. The domain of $$ f(x) $$ is all real $$ x $$.

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1. First, identify the domain by finding the values of $$ x $$ for which the function is defined. The function is given by

$$
f(x)=\frac{9x}{1-x^2}.
$$

2. The function is undefined when the denominator is zero. Set the denominator equal to zero:

$$
1-x^2=0.
$$

3. Solve for $$ x $$:

$$
x^2=1 \quad \Rightarrow \quad x=1 \quad \text{or} \quad x=-1.
$$

4. Therefore, the domain of $$ f(x) $$ is all real numbers except when $$ x=1 $$ or $$ x=-1 $$. In answer choice form, the correct selection is:

**A. The domain of $$ f(x) $$ is all real $$ x $$, except $$ x=-1,1 $$.**

5. Additional pertinent information from the graphing strategy:
   - **Vertical asymptotes:** Occur when $$ x=-1 $$ and $$ x=1 $$ (since the function is undefined at these $$ x $$-values).
   - **Horizontal asymptote:** As $$ x \to \pm \infty $$,
   
     $$
     f(x) \sim \frac{9x}{-x^2}=-\frac{9}{x}\to 0,
     $$
     
     so the horizontal asymptote is $$ y=0 $$.
   - **Intercepts:**
     - The $$ x $$-intercept is at $$ x=0 $$ (since $$ f(0)=0 $$).
     - The $$ y $$-intercept is also $$ 0 $$ (when $$ x=0 $$).
   - **Symmetry:** The function is odd because
      
     $$
     f(-x)=\frac{-9x}{1-(-x)^2}=-\frac{9x}{1-x^2}=-f(x),
     $$
     
     which implies that the graph is symmetric with respect to the origin.

6. **Sketching the graph:**
   - Plot vertical asymptotes at $$ x=-1 $$ and $$ x=1 $$.
   - Plot the horizontal asymptote $$ y=0 $$.
   - Mark the intercept at the origin $$ (0,0) $$.
   - Use the symmetry about the origin to sketch the two sides of the graph accordingly.
   - Note that the behavior near the vertical asymptotes will show the function diverging to $$ \infty $$ or $$ -\infty $$ as $$ x $$ approaches $$-1$$ and $$ 1 $$ from the left and right.

The final correct answer is Choice A: The domain of $$ f(x) $$ is all real $$ x $$, except $$ x=-1,1 $$.
The domain of $$ f(x) $$ is all real numbers except $$ x=-1 $$ and $$ x=1 $$.

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