https://file.aiquickdraw.com/mathbot/file/1729578195536tmljxeh6.jpg

The problem in the image is asking to find \( (f \circ g \circ h)(x) \), which is the composition of three functions:

- \( f(x) = \sqrt{x} \)
- \( g(x) = \frac{1}{x} \)
- \( h(x) = x^3 \)

To solve this, we will perform function composition step by step.

### Step 1: Find \( (g \circ h)(x) \)
This means applying \( h(x) \) first and then applying \( g \) to the result. So:

\[
g(h(x)) = g(x^3) = \frac{1}{x^3}
\]

### Step 2: Find \( (f \circ (g \circ h))(x) \)
Now, apply \( f \) to the result from the previous step:

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

So, the final result is:

\[
(f \circ g \circ h)(x) = \frac{1}{x^{3/2}}
\]

Would you like further details or explanations on any step?

### Related Questions:
1. How would the composition change if \( g(x) = \ln(x) \)?
2. What happens if we reverse the order of composition to \( (h \circ g \circ f)(x) \)?
3. Can you simplify the result if \( h(x) = x^2 \) instead of \( x^3 \)?
4. How does the chain rule apply when differentiating composed functions like this?
5. What would be the inverse of the final composition \( f \circ g \circ h \)?

### Tip:
When doing compositions, always work from the innermost function outward!

Find (f ∘ g ∘ h)(x) if f(x) = √x, g(x) = 1/x, h(x) = x^3.

Learn how to solve the composition of functions (f ∘ g ∘ h)(x) where f(x) = √x, g(x) = 1/x, and h(x) = x^3. This problem explores key concepts in function composition, radicals, and rational expressions. Suitable for high school students studying Pre-Calculus or Calculus.

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