To find \(g[f(x)]\), you need to substitute the function \(f(x)\) into the function \(g(x)\).

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

To find \(g[f(x)]\), substitute \(f(x)\) into \(g(x)\):

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

Now, replace \(x\) in \(g(x)\) with \(x^2 + \sqrt{x}\):

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

Let's simplify each part:

1. **Numerator:**
   \[
   (x^2 + \sqrt{x})^2 = (x^2 + \sqrt{x})(x^2 + \sqrt{x}) = x^4 + 2x^2\sqrt{x} + x
   \]

2. **

Explore how to compose functions g(x) and f(x) with an example. Learn step-by-step how to find g(f(x)) using algebraic substitutions. Suitable for advanced high school students and college-level learners.

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