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The question is asking to determine the composition of functions \((h \circ g \circ f)(x)\) and \((f \circ g \circ h)(x)\), given the functions:

\[
f(x) = 3x + 1, \quad g(x) = \frac{1}{x - 1}, \quad \text{and} \quad h(x) = x^2 - 4.
\]

Let's solve these step by step.

### 1. Finding \((h \circ g \circ f)(x)\):
This is the composition of three functions, meaning we need to substitute one into the other in the specified order.

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

#### Step 1: Find \(f(x)\):
\[
f(x) = 3x + 1
\]

#### Step 2: Find \(g(f(x))\):
Substitute \(f(x)\) into \(g(x)\):
\[
g(f(x)) = g(3x + 1) = \frac{1}{(3x + 1) - 1} = \frac{1}{3x}
\]

#### Step 3: Find \(h(g(f(x)))\):
Substitute \(g(f(x))\) into \(h(x)\):
\[
h(g(f(x))) = h\left(\frac{1}{3x}\right) = \left(\frac{1}{3x}\right)^2 - 4 = \frac{1}{9x^2} - 4
\]

So,
\[
(h \circ g \circ f)(x) = \frac{1}{9x^2} - 4
\]

### 2. Finding \((f \circ g \circ h)(x)\):
This is another composition in a different order:

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

#### Step 1: Find \(h(x)\):
\[
h(x) = x^2 - 4
\]

#### Step 2: Find \(g(h(x))\):
Substitute \(h(x)\) into \(g(x)\):
\[
g(h(x)) = g(x^2 - 4) = \frac{1}{(x^2 - 4) - 1} = \frac{1}{x^2 - 5}
\]

#### Step 3: Find \(f(g(h(x)))\):
Substitute \(g(h(x))\) into \(f(x)\):
\[
f(g(h(x))) = f\left(\frac{1}{x^2 - 5}\right) = 3\left(\frac{1}{x^2 - 5}\right) + 1 = \frac{3}{x^2 - 5} + 1
\]

So,
\[
(f \circ g \circ h)(x) = \frac{3}{x^2 - 5} + 1
\]

These are the final results:

- \((h \circ g \circ f)(x) = \frac{1}{9x^2} - 4\)
- \((f \circ g \circ h)(x) = \frac{3}{x^2 - 5} + 1\)

Would you like any further details or have any questions?

### Related Questions:
1. What is the general method for finding the composition of three functions?
2. How do you interpret the result of a composition of functions graphically?
3. What are the domain restrictions for the compositions \((h \circ g \circ f)(x)\) and \((f \circ g \circ h)(x)\)?
4. How would the composition change if the functions were different?
5. Can you decompose a complex function into simpler component functions?

### Tip:
When performing function composition, always start with the innermost function and work your way outwards step by step. This systematic approach helps avoid mistakes.

Jika f(x) = 3x + 1, g(x) = 1/(x - 1), dan h(x) = x^2 - 4, tentukan (h ∘ g ∘ f)(x) dan (f ∘ g ∘ h)(x)!

Learn how to solve function compositions involving f(x) = 3x + 1, g(x) = 1/(x - 1), and h(x) = x^2 - 4. Detailed steps to find (h ∘ g ∘ f)(x) and (f ∘ g ∘ h)(x) with algebraic techniques. Suitable for high school students.

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