To find \(\left(\frac{f}{g}\right)(x)\), we need to divide the function \(f(x)\) by \(g(x)\).

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

Now, \(\left(\frac{f}{g}\right)(x)\) is calculated as:

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

To simplify, multiply the numerator and the denominator by \(x-1\):

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

Now, distribute \(x-1\) in the numerator:

\[
\left(\frac{f}{g}\right)(x) = \frac{x^2(x-1) + \sqrt{x}(x-1)}{x^2} = \frac{x^3 - x^2 + \sqrt{x}(x-1)}{x^2}
\]

So, the final expression is:

\[
\left(\frac{f}{g}\right)(x) = \frac{x^3 - x^2 + \sqrt{x}(x-1)}{x^2}
\]

Would you like to see the steps in more detail or have any questions?

Here are five related questions:
1. How do you simplify complex rational expressions?
2. What are the restrictions on \(x\) for the function \(g(x)\)?
3. How would you find the domain of \(\left(\frac{f}{g}\right)(x)\)?
4. What happens to \(\left(\frac{f}{g}\right)(x)\) as \(x\) approaches 1?
5. Can \(\sqrt{x}\) be expressed in another form for simplification?

**Tip:** Always check for any restrictions on the variable when working with rational functions to ensure the function is defined.

Explore how to calculate and simplify the composite function (f/g)(x) where f(x) = x^2 + √x and g(x) = x^2 / (x - 1). This detailed solution covers the steps involved in dividing these functions and understanding the resulting rational function. Suitable for advanced high school students and anyone interested in algebraic function composition.

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