To find the hole in the given rational function:

$\frac{x^2 - x - 2}{x^2 - 1}$

Step 1: Factor the Numerator and Denominator

Factor both the numerator and denominator:

• Numerator: $x^2 - x - 2 = (x - 2)(x + 1)$
• Denominator: $x^2 - 1 = (x - 1)(x + 1)$

Now, the function becomes:

$\frac{(x - 2)(x + 1)}{(x - 1)(x + 1)}$

Step 2: Identify Common Factors

The common factor in both the numerator and denominator is $(x + 1)$. Cancel this factor:

$\frac{x - 2}{x - 1}, \quad x \neq -1$

Since $x = -1$ was canceled, there is a hole in the function at $x = -1$.

Step 3: Find the y-Coordinate of the Hole

To determine the y-coordinate, substitute $x = -1$ into the simplified function:

$y = \frac{-1 - 2}{-1 - 1} = \frac{-3}{-2} = \frac{3}{2}$

Final Answer:

The hole is at $(-1, \frac{3}{2})$.

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Would you like a step-by-step graphical explanation?

Related Questions:

1. How do you determine vertical and horizontal asymptotes for rational functions?
2. What happens when there are no common factors in a rational function?
3. How do you find intercepts for rational functions?
4. What are removable discontinuities in rational functions?
5. How does graphing help visualize holes in functions?

Tip:

When finding holes, always fully factor the function first to check for common factors!