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.