Natching graphs whth rational functions Two wertical asymptotes Consider the following rational functions, $$ g(x)=\frac{2 x-4}{x^{2}-2 x-8} $$ $$ h(x)=\frac{x^{2}+x-6}{x^{2}-x-6} $$

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To analyze the rational functions $$ g(x) $$ and $$ h(x) $$ and find their vertical asymptotes, we need to follow these steps:

1. **Identify the vertical asymptotes**: Vertical asymptotes occur where the denominator of the rational function is equal to zero, provided that the numerator is not also zero at those points.

2. **Factor the denominators**: This will help us find the values of $$ x $$ that make the denominator zero.

Let's start with the function $$ g(x) $$:

### Step 1: Analyze $$ g(x) $$

The function is given by:
$$
g(x) = \frac{2x - 4}{x^2 - 2x - 8}
$$

**Factor the denominator**:
We need to factor $$ x^2 - 2x - 8 $$.

### Step 2: Factor $$ x^2 - 2x - 8 $$

To factor $$ x^2 - 2x - 8 $$, we look for two numbers that multiply to $$-8$$ and add to $$-2$$. The numbers $$-4$$ and $$2$$ work.

Thus, we can factor it as:
$$
x^2 - 2x - 8 = (x - 4)(x + 2)
$$

Now, we can rewrite $$ g(x) $$:
$$
g(x) = \frac{2x - 4}{(x - 4)(x + 2)}
$$

**Find vertical asymptotes**:
Set the denominator equal to zero:
$$
(x - 4)(x + 2) = 0
$$
This gives us:
$$
x - 4 = 0 \quad \Rightarrow \quad x = 4
$$
$$
x + 2 = 0 \quad \Rightarrow \quad x = -2
$$

So, the vertical asymptotes for $$ g(x) $$ are at $$ x = 4 $$ and $$ x = -2 $$.

### Step 3: Analyze $$ h(x) $$

The function is given by:
$$
h(x) = \frac{x^2 + x - 6}{x^2 - x - 6}
$$

**Factor the denominator**:
We need to factor $$ x^2 - x - 6 $$.

### Step 4: Factor $$ x^2 - x - 6 $$

To factor $$ x^2 - x - 6 $$, we look for two numbers that multiply to $$-6$$ and add to $$-1$$. The numbers $$-3$$ and $$2$$ work.

Thus, we can factor it as:
$$
x^2 - x - 6 = (x - 3)(x + 2)
$$

Now, we can rewrite $$ h(x) $$:
$$
h(x) = \frac{x^2 + x - 6}{(x - 3)(x + 2)}
$$

**Find vertical asymptotes**:
Set the denominator equal to zero:
$$
(x - 3)(x + 2) = 0
$$
This gives us:
$$
x - 3 = 0 \quad \Rightarrow \quad x = 3
$$
$$
x + 2 = 0 \quad \Rightarrow \quad x = -2
$$

So, the vertical asymptotes for $$ h(x) $$ are at $$ x = 3 $$ and $$ x = -2 $$.

### Summary of Vertical Asymptotes

- For $$ g(x) $$: Vertical asymptotes at $$ x = 4 $$ and $$ x = -2 $$.
- For $$ h(x) $$: Vertical asymptotes at $$ x = 3 $$ and $$ x = -2 $$.

Both functions share a vertical asymptote at $$ x = -2 $$.
The vertical asymptotes for $$ g(x) $$ are at $$ x = 4 $$ and $$ x = -2 $$. For $$ h(x) $$, the vertical asymptotes are at $$ x = 3 $$ and $$ x = -2 $$. Both functions share a vertical asymptote at $$ x = -2 $$.

Natching graphs whth rational functions Two wertical asymptotes Consider the following rational functions, \[ g(x)=\frac{2 x-4}{x^{2}-2 x-8} \] \( h(x)=\frac{x^{2}+x-6}{x^{2}-x-6} \)

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