1. **State the problem:** Find the domain and range of the inverse function of $f(x) = \frac{1}{x} - 2$. 2. **Recall the inverse function:** To find the inverse, swap $x$ and $y$ and solve for $y$: $$x = \frac{1}{y} - 2$$ 3. **Solve for $y$:** $$x + 2 = \frac{1}{y}$$ $$y = \frac{1}{x + 2}$$ 4. **Domain of the inverse:** The inverse function is $f^{-1}(x) = \frac{1}{x + 2}$. The domain is all real numbers except where the denominator is zero: $$x + 2 \neq 0 \implies x \neq -2$$ 5. **Range of the inverse:** The range of the inverse is the domain of the original function. The original function's domain is all real numbers except $x \neq 0$. 6. **Summary:** - Domain of $f^{-1}$: $\{x \in \mathbb{R} : x \neq -2\}$ - Range of $f^{-1}$: $\{y \in \mathbb{R} : y \neq 0\}$