Find the vertical asymptotes, if any, of the graph of the rational function. * $$ g(x)=\frac{x}{x^{2}-1} $$

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To find the vertical asymptotes of the rational function $$ g(x) = \frac{x}{x^2 - 1} $$, follow these steps:

1. **Identify Points Where the Denominator is Zero:**
   
   Vertical asymptotes occur where the denominator of the rational function is zero (since the function is undefined there) **and** the numerator is not zero.

Set the denominator equal to zero and solve for $$ x $$:
   $$
   x^2 - 1 = 0 \implies (x - 1)(x + 1) = 0 \implies x = 1 \text{ or } x = -1
   $$

2. **Check the Numerator at These Points:**
   
   - For $$ x = 1 $$:
     $$
     \text{Numerator} = 1 \neq 0
     $$
   - For $$ x = -1 $$:
     $$
     \text{Numerator} = -1 \neq 0
     $$
   
   Since the numerator is not zero at either $$ x = 1 $$ or $$ x = -1 $$, both points are indeed vertical asymptotes.

3. **Conclusion:**
   
   The vertical asymptotes of the function $$ g(x) = \frac{x}{x^2 - 1} $$ are the lines:
   $$
   x = 1 \quad \text{and} \quad x = -1
   $$

**Graphical Representation:**

Here's a rough sketch of the function highlighting the vertical asymptotes:

$$
\begin{array}{c}
\text{Vertical Asymptotes at } x = 1 \text{ and } x = -1 \
\begin{tikzpicture}[scale=1]
    \draw[->] (-3,0) -- (3,0) node[right] {$x$};
    \draw[->] (0,-3) -- (0,3) node[above] {$g(x)$};
    \draw[dashed] (1,-3) -- (1,3);
    \draw[dashed] (-1,-3) -- (-1,3);
    \draw[domain=-3:-1.1, smooth, variable=\x, blue] plot ({\x}, { \x/(\x*\x -1)});
    \draw[domain=-0.9:0.9, smooth, variable=\x, blue] plot ({\x}, { \x/(\x*\x -1)});
    \draw[domain=1.1:3, smooth, variable=\x, blue] plot ({\x}, { \x/(\x*\x -1)});
\end{tikzpicture}
\end{array}
$$

The dashed lines represent the vertical asymptotes.

**Final Answer:**

All vertical asymptotes of g are the lines x = 1 and x = –1.
The vertical asymptotes of the function $$ g(x) = \frac{x}{x^2 - 1} $$ are at $$ x = 1 $$ and $$ x = -1 $$.

Find the vertical asymptotes, if any, of the graph of the rational function.

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