1. The problem asks to determine the domain and range of the inverse function $y = f^{-1}(x)$ given the domain and range of $y = f(x)$. 2. Recall that the domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$. This is because the inverse function swaps the roles of inputs and outputs. 3. Given the domain of $f$ is $\{x \mid 1 \leq x \leq 2, x \in \mathbb{R}\}$, this means the range of $f^{-1}$ is $\{y \mid 1 \leq y \leq 2, y \in \mathbb{R}\}$. 4. Given the range of $f$ is $\{y \mid 3 \leq y \leq 4, y \in \mathbb{R}\}$, this means the domain of $f^{-1}$ is $\{x \mid 3 \leq x \leq 4, x \in \mathbb{R}\}$. 5. Therefore, the answers are: 1. Domain of $f^{-1}$ = $\{x \mid 3 \leq x \leq 4, x \in \mathbb{R}\}$ 2. Range of $f^{-1}$ = $\{y \mid 1 \leq y \leq 2, y \in \mathbb{R}\}$ 3. Range of $f^{-1}$ = $\{y \mid 1 \leq y \leq 2, y \in \mathbb{R}\}$ (same as 2) 4. Domain of $f^{-1}$ = $\{x \mid 3 \leq x \leq 4, x \in \mathbb{R}\}$ (same as 1)