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.

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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:

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

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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:

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

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3. Final expression of the function:

With the simplification, the function can be rewritten as:

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

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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.

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5. Summary:

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

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

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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.