1. **State the problem:** We are given the linear function $f(x) = -\frac{4}{3}x - 1$ and asked to find its domain. 2. **Recall the domain of linear functions:** The domain of a linear function is all real numbers because there are no restrictions such as division by zero or square roots of negative numbers. 3. **Apply this to the given function:** Since $f(x)$ is a linear function with no denominators involving $x$ and no radicals, its domain is all real numbers. 4. **Express the domain:** The domain is $\{x : x \in \mathbb{R}\}$. **Final answer:** The domain of $f(x) = -\frac{4}{3}x - 1$ is $\boxed{\{x : x \in \mathbb{R}\}}$.