Diketahui 𝑓(𝑥) = 3𝑥
2 − 𝑥 + 1, 𝑔(𝑥) = 𝑥 + 2, dan ℎ(𝑥) = 4𝑥 + 3 Tentukan :
a. (𝑓 ∘ 𝑔 ∘ ℎ)(𝑥)

To solve the problem of finding \((f \circ g \circ h)(x)\), we need to evaluate the composition of these three functions, which means applying one function to the result of another in a specific order.

Given:

- \( f(x) = 3x^2 - x + 1 \)
- \( g(x) = x + 2 \)
- \( h(x) = 4x + 3 \)

The expression \((f \circ g \circ h)(x)\) represents \(f(g(h(x)))\). We will evaluate this step-by-step:

### Step 1: Evaluate \( h(x) \)

\[ 
h(x) = 4x + 3 
\]

### Step 2: Evaluate \( g(h(x)) \)

Substitute \( h(x) \) into \( g(x) \):

\[
g(h(x)) = g(4x + 3) = (4x + 3) + 2 = 4x + 5
\]

### Step 3: Evaluate \( f(g(h(x))) \)

Substitute \( g(h(x)) \) into \( f(x) \):

\[
f(g(h(x))) = f(4x + 5) = 3(4x + 5)^2 - (4x + 5) + 1
\]

### Step 4: Simplify \( f(4x + 5) \)

First, expand \((4x + 5)^2\):

\[
(4x + 5)^2 = (4x)^2 + 2 \cdot 4x \cdot 5 + 5^2 = 16x^2 + 40x + 25
\]

Now, substitute and simplify:

\[
f(4x + 5) = 3(16x^2 + 40x + 25) - (4x + 5) + 1
\]

Distribute the 3:

\[
= 48x^2 + 120x + 75 - 4x - 5 + 1
\]

Combine like terms:

\[
= 48x^2 + 116x + 71
\]

Thus, the final result is:

\[
(f \circ g \circ h)(x) = 48x^2 + 116x + 71
\]

If you have any questions or would like further details, feel free to ask!

Here are some related questions for further exploration:

1. How do you compose functions in general?
2. What is the difference between \((f \circ g)(x)\) and \((g \circ f)(x)\)?
3. Can you evaluate \((g \circ f)(x)\) with the given functions?
4. How do you find the inverse of a function?
5. What are the properties of polynomial functions?
6. How can we determine the domain of a composed function?
7. How do you graph the composition of functions?
8. Can composition of functions be applied to trigonometric functions?

**Tip:** When composing functions, always start by evaluating the innermost function first and work outward.

Learn how to compose functions (f ∘ g ∘ h)(x) = 48x^2 + 116x + 71 step-by-step with detailed explanations. This problem involves function composition using polynomial functions and includes polynomial expansion techniques. Suitable for high school students.

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