The problem asks to find the compositions $(h \circ g \circ f)(x)$ and $(f \circ g \circ h)(x)$ given the following functions:

1. $f(x) = 3x + 1$
2. $g(x) = \frac{1}{x - 1}$
3. $h(x) = x^2 - 4$

Finding $(h \circ g \circ f)(x)$:

This represents applying $f(x)$, then applying $g(x)$, and finally applying $h(x)$.

$(h \circ g \circ f)(x) = h(g(f(x)))$

• First, apply $f(x)$: $f(x) = 3x + 1$
• Now, apply $g(x)$ to $f(x)$: $g(f(x)) = g(3x + 1) = \frac{1}{(3x + 1) - 1} = \frac{1}{3x}$
• Finally, apply $h(x)$ to $g(f(x))$: $h(g(f(x))) = h\left(\frac{1}{3x}\right) = \left(\frac{1}{3x}\right)^2 - 4 = \frac{1}{9x^2} - 4$

Thus, $(h \circ g \circ f)(x) = \frac{1}{9x^2} - 4$

Finding $(f \circ g \circ h)(x)$:

This represents applying $h(x)$, then applying $g(x)$, and finally applying $f(x)$.

$(f \circ g \circ h)(x) = f(g(h(x)))$

• First, apply $h(x)$: $h(x) = x^2 - 4$
• Now, apply $g(x)$ to $h(x)$: $g(h(x)) = g(x^2 - 4) = \frac{1}{(x^2 - 4) - 1} = \frac{1}{x^2 - 5}$
• Finally, apply $f(x)$ to $g(h(x))$: $f(g(h(x))) = f\left(\frac{1}{x^2 - 5}\right) = 3\left(\frac{1}{x^2 - 5}\right) + 1 = \frac{3}{x^2 - 5} + 1$

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

Final Answers:

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

Would you like any more details or further clarifications? Here are 5 related questions for further exploration:

1. How do you evaluate the domain of the function $(h \circ g \circ f)(x)$?
2. What are the domains of $g(x)$ and $h(x)$ separately?
3. How does function composition relate to real-world applications?
4. What happens if we swap the order of compositions, like $(g \circ f \circ h)(x)$?
5. Can you simplify the expressions further for specific values of $x$?

Tip: When composing functions, always apply them step-by-step, starting from the innermost function.