The function you've provided is \( f(x) = \sqrt{x^2 - 5x} \).

### Simplifying the Function:
The function involves a square root, which requires the expression inside the square root to be non-negative. This means that:

\[
x^2 - 5x \geq 0
\]

To solve this inequality, let's first solve the corresponding equation:

\[
x^2 - 5x = 0
\]

Factor the quadratic:

\[
x(x - 5) = 0
\]

So, the solutions are \( x = 0 \) and \( x = 5 \). These are the points where the expression inside the square root is zero. To determine where the expression is non-negative, we analyze the intervals:

1. **Interval \( (-\infty, 0) \):**
   \[
   x^2 - 5x > 0 \quad \text{(True, because \( x(x - 5) > 0 \) when \( x < 0 \))}
   \]
2. **Interval \( (0, 5) \):**
   \[
   x^2 - 5x < 0 \quad \text{(False, because \( x(x - 5) < 0 \) when \( 0 < x < 5 \))}
   \]
3. **Interval \( (5, \infty) \):**
   \[
   x^2 - 5x > 0 \quad \text{(True, because \( x(x - 5) > 0 \) when \( x > 5 \))}
   \]

### Domain of the Function:
From the above analysis, the expression \( x^2 - 5x \geq 0 \) is satisfied in the intervals \( (-\infty, 0] \) and \( [5, \infty) \). Therefore, the domain of the function \( f(x) = \sqrt{x^2 - 5x} \) is:

\[
(-\infty, 0] \cup [5, \infty)
\]

### Summary:
- The function \( f(x) = \sqrt{x^2 - 5x} \) is defined for \( x \in (-\infty, 0] \cup [5, \infty) \).
- The function involves square roots, so the expression inside must be non-negative, leading to the specified domain.

Would you like more details on solving the inequality, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How do you solve quadratic inequalities in general?
2. What are the steps to factor a quadratic expression?
3. How do you determine the domain of a function involving a square root?
4. Can you explain the significance of the critical points \( x = 0 \) and \( x = 5 \) in this problem?
5. What happens graphically when you have a function with a square root?

**Tip:** When dealing with square roots in functions, always check the domain by ensuring the expression inside the square root is non-negative.

Explore how to determine the domain of the function f(x) = sqrt(x^2 - 5x) by solving the quadratic inequality. This problem highlights key algebraic concepts like factoring quadratics and understanding the domain of square root functions. Ideal for students in Grades 9-11.

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