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Let's go through the questions one by one.

### 8. (a)
You are asked to find two compositions: \( (f \circ g)(x) \) and \( (g \circ f)(x) \).

- \( f(x) = x^{-1} = \frac{1}{x} \)
- \( g(x) = (x^2 - 1)^{-3} \)

**First, let's find \( (f \circ g)(x) \):**
This is the composition of \( f \) with \( g \), meaning \( f(g(x)) \). So, we substitute \( g(x) \) into \( f(x) \):

\[
(f \circ g)(x) = f(g(x)) = f((x^2 - 1)^{-3}) = \frac{1}{(x^2 - 1)^{-3}} = (x^2 - 1)^3
\]

**Next, let's find \( (g \circ f)(x) \):**
This is the composition of \( g \) with \( f \), meaning \( g(f(x)) \). So, we substitute \( f(x) \) into \( g(x) \):

\[
(g \circ f)(x) = g(f(x)) = g\left( \frac{1}{x} \right) = \left( \left( \frac{1}{x} \right)^2 - 1 \right)^{-3}
\]
Simplifying the expression inside the parentheses:
\[
\left( \frac{1}{x} \right)^2 = \frac{1}{x^2}, \quad \text{so} \quad \left( \frac{1}{x^2} - 1 \right)^{-3} = \left( \frac{1 - x^2}{x^2} \right)^{-3} = \left( \frac{x^2}{1 - x^2} \right)^3 = \frac{x^6}{(1 - x^2)^3}
\]

Thus, \( (g \circ f)(x) = \frac{x^6}{(1 - x^2)^3} \).

### 8. (b)
- \( f(x) = \sqrt{2 - x} \)
- \( g(x) = \sqrt{x - 2} \)

We are tasked with finding \( (g \circ f)(x) \), which is \( g(f(x)) \):

\[
(g \circ f)(x) = g(f(x)) = g(\sqrt{2 - x}) = \sqrt{\sqrt{2 - x} - 2}
\]
This simplifies to:
\[
(g \circ f)(x) = \sqrt{\sqrt{2 - x} - 2}
\]

There is no further simplification.

### 8. (c)
- \( f(x) = 12 - x^2 \)
- \( g(x) = 4 \)

We are asked to find both \( (f \circ g)(x) \) and \( (g \circ f)(x) \).

**First, \( (f \circ g)(x) \):**
This is \( f(g(x)) \), so substitute \( g(x) = 4 \) into \( f(x) \):

\[
(f \circ g)(x) = f(4) = 12 - 4^2 = 12 - 16 = -4
\]

**Next, \( (g \circ f)(x) \):**
This is \( g(f(x)) \), and since \( g(x) = 4 \) is constant, it does not depend on \( f(x) \). Therefore, for any input \( x \), \( (g \circ f)(x) = 4 \).

Thus, \( (g \circ f)(x) = 4 \).

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Would you like to dive deeper into any part of the solution, or have any further questions?

Here are 5 related questions:
1. How do you generally simplify compositions of functions involving powers?
2. Can you explain how domain restrictions affect the compositions of functions?
3. How would you verify if a composition of functions is defined for all real numbers?
4. What happens if the composition involves a function with square roots of negative numbers?
5. How do you interpret the inverse function when composing two functions?

**Tip**: Always ensure

Let f(x) = x^-1 and g(x) = (x^2 - 1)^-3. Express the compositions (f o g)(x) and (g o f)(x) as fractions. Let f(x) = √(2 - x) and g(x) = √(x - 2). Find the composition (g o f)(x). Let f(x) = 12 - x^2 and g(x) = 4. Find the compositions (f o g)(x) and (g o f)(x).

This problem explores how to express the compositions of functions like f(x) = x^-1 and g(x) = (x^2 - 1)^-3, and square root functions, through detailed step-by-step solutions.

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