Find 
f ∘ g ∘ h.
f(x) = 4x − 5,    g(x) = sin(x),    h(x) = x2
(f ∘ g ∘ h)(x) = 
 

To find \((f \circ g \circ h)(x)\), we need to apply the composition of functions step by step. This notation means:

\[
(f \circ g \circ h)(x) = f(g(h(x)))
\]

### Step 1: Apply \( h(x) \)
Given \( h(x) = x^2 \), we replace \( h(x) \) in the expression:
\[
g(h(x)) = g(x^2)
\]

### Step 2: Apply \( g(x) \)
Given \( g(x) = \sin(x) \), we replace \( g(x^2) \):
\[
g(x^2) = \sin(x^2)
\]

### Step 3: Apply \( f(x) \)
Given \( f(x) = 4x - 5 \), we replace \( f(g(x^2)) \) with \( f(\sin(x^2)) \):
\[
f(\sin(x^2)) = 4\sin(x^2) - 5
\]

### Final Answer:
\[
(f \circ g \circ h)(x) = 4\sin(x^2) - 5
\]

Would you like more details on function composition or any other clarification?

Here are 5 related questions to expand on this:
1. How does function composition work in general?
2. What are the domains and ranges of the individual functions?
3. How would the result change if \( h(x) \) were a different function?
4. Can you find the derivative of \( (f \circ g \circ h)(x) \)?
5. How would you evaluate \( (f \circ g \circ h)(\pi) \)?

**Tip:** When composing functions, work from the innermost function outward, applying each step sequentially.

Find f ∘ g ∘ h. f(x) = 4x − 5, g(x) = sin(x), h(x) = x^2 (f ∘ g ∘ h)(x) =

Learn how to solve the composition of functions (f ∘ g ∘ h)(x) where f(x) = 4x - 5, g(x) = sin(x), and h(x) = x^2. This problem covers function composition, trigonometry, and quadratic functions, with a detailed step-by-step explanation. Suitable for students in Grades 10-12.

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