8. Find the coordinates of the hole in the graph of the rational function \( f(x)=\frac{x^{2}-2 x-3}{x^{2}-x-6} \)

To find the coordinates of the hole in the graph of the rational function \( f(x) = \frac{x^2 - 2x - 3}{x^2 - x - 6} \), we first need to factor both the numerator and the denominator. 1. **Factor the numerator**: \[ x^2 - 2x - 3 = (x - 3)(x + 1) \] 2. **Factor the denominator**: \[ x^2 - x - 6 = (x - 3)(x + 2) \] Now we can rewrite the function: \[ f(x) = \frac{(x - 3)(x + 1)}{(x - 3)(x + 2)} \] 3. **Identify the hole**: The factor \( (x - 3) \) is present in both the numerator and the denominator. This indicates that there is a hole in the graph at \( x = 3 \). 4. **Find the y-coordinate of the hole**: To find the y-coordinate of the hole, we need to evaluate \( f(x) \) at \( x = 3 \), but we must first simplify the function by canceling the common factor: \[ f(x) = \frac{x + 1}{x + 2} \quad \text{for } x \neq 3 \] Now, we can substitute \( x = 3 \) into the simplified function: \[ f(3) = \frac{3 + 1}{3 + 2} = \frac{4}{5} \] 5. **Coordinates of the hole**: Thus, the coordinates of the hole in the graph of the function are: \[ \boxed{(3, \frac{4}{5})} \]