To determine the correct pairing of the domain restriction for $f(x)$ and its corresponding impact on $f^{-1}(x)$, recall the following key points about a function and its inverse:

1. The domain of $f(x)$ corresponds to the range of $f^{-1}(x)$.
2. The range of $f(x)$ corresponds to the domain of $f^{-1}(x)$.

Let’s analyze each given option:

1. $f(x)$ domain: $x \geq 1$, $f^{-1}(x)$ domain: $x \geq 1$
   This is incorrect because the domain of $f(x)$ should correspond to the range of $f^{-1}(x)$, not its domain.

2. $f(x)$ domain: $x \geq 1$, $f^{-1}(x)$ range: $y \geq 1$
   This is correct because the domain restriction on $f(x)$ directly affects the range of its inverse.

3. $f(x)$ domain: $x \leq -2$, $f^{-1}(x)$ domain: $x \leq -2$
   This is incorrect for the same reason as option 1.

4. $f(x)$ domain: $x \leq -2$, $f^{-1}(x)$ range: $y \leq -2$
   This is correct because restricting the domain of $f(x)$ to $x \leq -2$ results in the range of $f^{-1}(x)$ being $y \leq -2$.

Correct pairings:

• Option 2: $f(x)$ domain: $x \geq 1$, $f^{-1}(x)$ range: $y \geq 1$
• Option 4: $f(x)$ domain: $x \leq -2$, $f^{-1}(x)$ range: $y \leq -2$

Would you like further clarification on any part? Here are related questions:

1. How do you find the inverse of a function?
2. Why do domain and range switch in inverses?
3. What are the conditions for a function to have an inverse?
4. How do you graph a function and its inverse?
5. What are common domain restrictions in real-world applications?

Tip: When working with inverses, always remember: domain of $f(x)$ ↔ range of $f^{-1}(x)$.