Find the implied domain of the given function. $$ f(x)=\frac{x+13}{x^{2}-1 x-6} $$

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1. The function is given by  
   $$
   f(x)=\frac{x+13}{x^{2} - x - 6}.
   $$
   
2. The domain of $$ f(x) $$ consists of all values of $$ x $$ for which the function is defined. For rational functions, $$ f(x) $$ is undefined when the denominator equals zero.

3. Set the denominator equal to zero:  
   $$
   x^{2} - x - 6 = 0.
   $$

4. Factor the quadratic equation. We look for two numbers that multiply to $$-6$$ and add to $$-1$$. The numbers $$-3$$ and $$2$$ satisfy these conditions because:
   $$
   -3 \times 2 = -6 \quad \text{and} \quad -3 + 2 = -1.
   $$
   So the quadratic factors as:
   $$
   (x - 3)(x + 2) = 0.
   $$

5. Solve for $$ x $$ by setting each factor equal to zero:
   
   - $$ x - 3 = 0 $$ gives $$ x = 3 $$.
   - $$ x + 2 = 0 $$ gives $$ x = -2 $$.

6. The values $$ x = 3 $$ and $$ x = -2 $$ make the denominator zero, so these must be excluded from the domain.

7. Hence, the implied domain of $$ f(x) $$ is all real numbers except $$ x = 3 $$ and $$ x = -2 $$:
   $$
   \text{Domain: } \{x \in \mathbb{R} \mid x \neq 3 \text{ and } x \neq -2\}.
   $$
The domain of $$ f(x) $$ is all real numbers except $$ x = 3 $$ and $$ x = -2 $$.

Find the implied domain of the given function. \( f(x)=\frac{x+13}{x^{2}-1 x-6} \)

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