The function given is:

\[ f(x) = \sqrt{x^2 - x - 6} \]

To analyze this function, let's consider the domain and any potential issues with the square root.

### Step 1: Domain of the Function

The expression inside the square root, \( x^2 - x - 6 \), must be non-negative for the square root to be real. So, we set up the inequality:

\[ x^2 - x - 6 \geq 0 \]

### Step 2: Solve the Quadratic Inequality

First, solve the quadratic equation:

\[ x^2 - x - 6 = 0 \]

This can be factored as:

\[ (x - 3)(x + 2) = 0 \]

So, the roots are \( x = 3 \) and \( x = -2 \).

Now, we test the intervals determined by these roots:

1. For \( x < -2 \): Choose \( x = -3 \),
   \[ (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6 \geq 0 \]
   
2. For \( -2 < x < 3 \): Choose \( x = 0 \),
   \[ 0^2 - 0 - 6 = -6 \, (\text{which is negative}) \]
   
3. For \( x > 3 \): Choose \( x = 4 \),
   \[ 4^2 - 4 - 6 = 16 - 4 - 6 = 6 \geq 0 \]

Thus, the quadratic is non-negative when \( x \leq -2 \) or \( x \geq 3 \).

### Step 3: Conclusion

The domain of \( f(x) \) is:

\[ (-\infty, -2] \cup [3, \infty) \]

For these values of \( x \), the function \( f(x) \) is defined and real.

Would you like to explore further details, such as the range, or do you have any specific questions?

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1. What is the range of the function \( f(x) \)?
2. How does the function behave as \( x \) approaches the boundaries of its domain?
3. Can the function \( f(x) \) ever be zero?
4. How would the graph of \( f(x) \) look over its domain?
5. What happens if we shift the entire function by adding a constant \( c \)?

**Tip:** Always check the domain of a function when dealing with square roots to ensure you're only considering real-valued outputs.

Learn how to find the domain of the function f(x) = √x² - x - 6 by solving a quadratic inequality. This problem covers key concepts in algebra, including quadratic inequalities and square root functions. Ideal for students in Grades 9-11.

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