The function you provided is:

\[
f(x) = \sqrt{\frac{x^2 - x}{x - 1}}
\]

Let's analyze this function and determine its behavior in terms of domain and any simplifications.

---

### 1. **Simplifying the expression under the square root:**

First, simplify the fraction \(\frac{x^2 - x}{x - 1}\). Notice that the numerator \(x^2 - x\) can be factored:

\[
x^2 - x = x(x - 1)
\]

Thus:

\[
\frac{x^2 - x}{x - 1} = \frac{x(x - 1)}{x - 1}.
\]

For \(x \neq 1\), we can cancel the common factor \((x - 1)\), leaving:

\[
\frac{x^2 - x}{x - 1} = x \quad (\text{for } x \neq 1).
\]

Therefore, the function becomes:

\[
f(x) = \sqrt{x}, \quad \text{for } x \neq 1.
\]

---

### 2. **Domain analysis:**

Since we are dealing with a square root function, the expression inside the square root (which simplifies to \(x\)) must be non-negative:

\[
x \geq 0.
\]

Additionally, we must consider that the original fraction \(\frac{x^2 - x}{x - 1}\) is undefined at \(x = 1\). Therefore, the final domain is:

\[
x \geq 0 \quad \text{and} \quad x \neq 1.
\]

Thus, the domain of \(f(x)\) is:

\[
\text{Domain: } [0, 1) \cup (1, \infty).
\]

---

### 3. **Final expression of the function:**

With the simplification, the function can be rewritten as:

\[
f(x) = \sqrt{x}, \quad \text{for } x \neq 1.
\]

---

### 4. **Key Points about \(f(x)\):**

- **At \(x = 0\)**: \(f(0) = \sqrt{0} = 0\).
- **As \(x \to \infty\)**: \(f(x) \to \infty\).
- **At \(x = 1\)**: The function is undefined because of the original expression's denominator being zero.

---

### 5. **Summary:**

The simplified function is \(f(x) = \sqrt{x}\), with the domain \(x \geq 0\) but excluding \(x = 1\).

---

Would you like further clarification or details about graphing this function, continuity, or related properties?

---

Here are five follow-up questions:

1. How do we find the range of the function \(f(x) = \sqrt{x}\)?
2. What happens if the function had a cube root instead of a square root?
3. How would the graph of \(f(x) = \sqrt{\frac{x^2 - x}{x - 1}}\) look?
4. Can we determine if the function is continuous on its domain?
5. How would adding a constant inside or outside the square root affect the domain?

**Tip:** Always double-check for restrictions when dealing with rational functions and square roots to ensure proper domain analysis.

Analyze and simplify the function f(x) = \sqrt{\frac{x^2 - x}{x - 1}}, determine the function's domain, and explore key algebraic concepts like square roots and rational expressions. This solution is ideal for students in Grades 11-12.

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