For a function $f(x)$ and its inverse $f^{-1}(x)$, the domain of the inverse function is the same as the range of the original function, and the range of the inverse function is the same as the domain of the original function.

Given the following information for the original function $f(x)$:

• The domain of $f(x)$ is $(-\infty, 6)$
• The range of $f(x)$ is $(-5, \infty)$

Part a: Domain of $f^{-1}(x)$

The domain of the inverse function is the range of the original function, which is $(-5, \infty)$.

So the correct answer is:

• Domain of $f^{-1}(x)$: $(-5, \infty)$

Part b: Range of $f^{-1}(x)$

The range of the inverse function is the domain of the original function, which is $(-\infty, 6)$.

So the correct answer is:

• Range of $f^{-1}(x)$: $(-\infty, 6)$

Would you like more details or have any questions?

Here are 5 related questions you could explore:

1. How is the inverse of a function found algebraically?
2. What are the graphical relationships between a function and its inverse?
3. Can all functions have inverses? Why or why not?
4. What is the domain and range of a composite function $f(f^{-1}(x))$?
5. How do you determine if a function is one-to-one and therefore has an inverse?

Tip: The graph of a function and its inverse is symmetric along the line $y = x$.