For a function $g$, the domain of its inverse function $g^{-1}$ is the range of the original function $g$, and the range of $g^{-1}$ is the domain of $g$.

Given:

• Domain of $g$: $(-\infty, 0]$
• Range of $g$: $[4, \infty)$

The domain of $g^{-1}$ will be the range of $g$, which is $[4, \infty)$.

Final Answer:

The domain of $g^{-1}$ is $[4, \infty)$.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How do you find the range of an inverse function?
2. What are the conditions for a function to have an inverse?
3. How does the graphical representation of a function relate to its inverse?
4. Can a function with a restricted domain have an inverse?
5. What does it mean for a function to be one-to-one?

Tip: A function is one-to-one if it passes both the horizontal and vertical line tests, meaning each $y$-value corresponds to exactly one $x$-value.