To find the domain of the inverse of a function, we should first consider the range of the original function, as the range of the function becomes the domain of its inverse.

Looking at the provided graph:

1. The function approaches positive infinity as $x \to 0^+$ (from the right side of the y-axis).
2. As $x \to -\infty$, the function value flattens out, approaching $y = 0$.
3. The graph indicates that the function values cover all $y$-values from $-\infty$ to $0$ (exclusive) and from $0$ to $+\infty$ (exclusive).

Therefore, the range of the function is: $(-\infty, 0) \cup (0, +\infty)$

Since this is the range of the function, it will be the domain of the inverse of the function.

Answer:

The domain of the inverse of the function is: $(-\infty, 0) \cup (0, +\infty)$

Would you like further details or explanations on any part of this solution?

Here are some related questions for further exploration:

1. What would be the range of the inverse of this function?
2. How do we determine the range of a function if only the graph is given?
3. What are the conditions for a function to have an inverse?
4. How does the vertical asymptote of a function affect the domain of its inverse?
5. Can the inverse of this function be represented as a function itself?

Tip: For any function, remember that the domain of its inverse is equivalent to the range of the original function.