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.