Find 
f ∘ g ∘ h.
f(x) = 9x − 2,    g(x) = sin(x),    h(x) = x2


To find \( f \circ g \circ h \), which is the composition of the functions \( f(x) \), \( g(x) \), and \( h(x) \), we'll follow these steps:

1. **Find \( h(x) \)**: This is the innermost function.
   \[
   h(x) = x^2
   \]

2. **Find \( g(h(x)) \)**: Apply \( g(x) \) to \( h(x) \).
   \[
   g(h(x)) = g(x^2) = \sin(x^2)
   \]

3. **Find \( f(g(h(x))) \)**: Finally, apply \( f(x) \) to \( g(h(x)) \).
   \[
   f(g(h(x))) = f(\sin(x^2)) = 9\sin(x^2) - 2
   \]

Thus, the composition \( f \circ g \circ h \) is given by:

\[
f \circ g \circ h(x) = 9\sin(x^2) - 2
\]

Would you like more details on any part of this solution?

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Here are 5 related questions for further practice:

1. What is \( g \circ h \) if \( g(x) = \cos(x) \) and \( h(x) = x + 1 \)?
2. Find \( f \circ g \) if \( f(x) = 3x + 4 \) and \( g(x) = x^3 - 2 \).
3. Compute \( f(x) \) given \( f(g(h(x))) = x^2 + 7 \) and \( g(x) = \sqrt{x} \), \( h(x) = 2x \).
4. What is the derivative of \( f \circ g \circ h(x) \) with respect to \( x \) given the functions in the problem?
5. If \( g(x) = \tan(x) \) and \( h(x) = \ln(x) \), what is \( g \circ h(x) \)?

**Tip:** When composing functions, always start from the innermost function and work your way outward.

Explore how to compute the composition f ∘ g ∘ h(x) = 9sin(x^2) - 2 step-by-step. This problem involves function composition with trigonometric and quadratic functions, illustrating advanced high school level concepts in mathematics.

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