Find f ∘ g ∘ h.
f(x) = square root x − 1
,    g(x) = x2,    h(x) = x3 + 2

To find \( f \circ g \circ h \), you need to evaluate the composition of functions \( f(g(h(x))) \). Let's break it down step by step:

### Step 1: Evaluate \( h(x) \)
Given:
\[ h(x) = x^3 + 2 \]

### Step 2: Evaluate \( g(h(x)) \)
Now, substitute \( h(x) \) into \( g(x) \):
\[ g(h(x)) = g(x^3 + 2) \]

Since \( g(x) = x^2 \), this becomes:
\[ g(h(x)) = (x^3 + 2)^2 \]

### Step 3: Evaluate \( f(g(h(x))) \)
Now, substitute \( g(h(x)) \) into \( f(x) \):
\[ f(g(h(x))) = f((x^3 + 2)^2) \]

Since \( f(x) = \sqrt{x} - 1 \), this becomes:
\[ f(g(h(x))) = \sqrt{(x^3 + 2)^2} - 1 \]

### Simplifying
The square root of a square simplifies to the absolute value of the expression inside:
\[ f(g(h(x))) = |x^3 + 2| - 1 \]

Therefore, the composition \( f \circ g \circ h \) is:
\[
f(g(h(x))) = |x^3 + 2| - 1
\]

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How do you find the composition of more than three functions?
2. What is the difference between \( f \circ g \) and \( g \circ f \)?
3. Can \( f \circ g \) be the same as \( g \circ f \)? Under what conditions?
4. How does the domain of \( f(x) \) affect the composition \( f(g(x)) \)?
5. What is the role of the absolute value in simplifying square root expressions?

**Tip:** Always check if a square root expression can be simplified further by considering the domain of the function.

Learn how to find the composition of functions f ∘ g ∘ h using the square root function, quadratic function, and cubic function. This problem demonstrates the steps involved in function composition, including the evaluation of nested functions. Suitable for high school students studying advanced algebra and function theory.

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