What is the domain of the inverse function *h^*−1(*x*)?

To find the domain of the inverse function \( h^{-1}(x) \), we need to understand the relationship between the function \( h(x) \) and its inverse.

### Steps to Determine the Domain of \( h^{-1}(x) \):

1. **Understand the Function \( h(x) \):** The domain of the inverse function \( h^{-1}(x) \) is determined by the range of the original function \( h(x) \). Specifically, the domain of \( h^{-1}(x) \) is the set of all \( y \)-values such that there is an \( x \) in the domain of \( h(x) \) for which \( h(x) = y \).

2. **Find the Range of \( h(x) \):** 
   - The range of \( h(x) \) consists of all possible outputs of the function.
   - This means that the domain of \( h^{-1}(x) \) is exactly the range of \( h(x) \).

3. **Domain of \( h^{-1}(x) \):**
   - To find the domain of \( h^{-1}(x) \), determine the range of \( h(x) \) and use that as the domain for \( h^{-1}(x) \).

### Example

- If \( h(x) \) is a simple function like \( h(x) = x^2 \), and assuming \( h(x) \) is defined on \( x \geq 0 \), the range of \( h(x) \) would be \( [0, \infty) \).
- Therefore, the domain of \( h^{-1}(x) \) (which is \( \sqrt{x} \) in this case) would also be \( [0, \infty) \).

In summary, **the domain of the inverse function \( h^{-1}(x) \) is the range of the original function \( h(x) \).**

Would you like details on how to find the range for a specific type of function?

### Related Questions:

1. What are the steps to find the range of a function?
2. How do you determine if a function is one-to-one?
3. What is the significance of a function being bijective in relation to its inverse?
4. How do you graphically interpret the inverse of a function?
5. What happens to the domain and range when you restrict a function?
6. Can a function have an inverse if it is not one-to-one?
7. How does the vertical line test relate to finding an inverse function?
8. What are common mistakes when finding the domain of an inverse function?

**Tip:** Always ensure a function is one-to-one before finding its inverse, as only one-to-one functions have inverses that are also functions.

Explore how to determine the domain of the inverse function h^-1(x) based on the relationship with the original function h(x). This detailed explanation covers the essential steps and examples, ensuring clarity in understanding inverse functions in mathematics.

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