To determine the domain and range of the inverse function \( h^{-1}(x) \) given the domain and range of \( h(x) \), we need to understand the relationship between a function and its inverse.
For a function \( h(x) \), the domain is the set of all possible input values (\( x \)) for which the function is defined, and the range is the set of all possible output values (\( y \)) that the function can produce. These can be denoted as:
- Domain of \( h(x) \): \( x < 5 \)
- Range of \( h(x) \): \( y < 4 \)
When we deal with the inverse function \( h^{-1}(x) \), the domain and range of the inverse function swap places with the range and domain of the original function respectively. This means:
- The domain of \( h^{-1}(x) \) will be the range of \( h(x) \).
- The range of \( h^{-1}(x) \) will be the domain of \( h(x) \).
Given the domain and range of \( h(x) \) as:
- Domain of \( h(x) \): \( x < 5 \)
- Range of \( h(x) \): \( y < 4 \)
We can now determine the domain and range of \( h^{-1}(x) \):
- The domain of \( h^{-1}(x) \) is the range of \( h(x) \), which is \( x < 4 \).
- The range of \( h^{-1}(x) \) is the domain of \( h(x) \), which is \( y < 5 \).
Therefore, the correct choice is:
D.
- Domain: \( x < 4 \)
- Range: \( y < 5 \)
Thus, the domain and range of the inverse function \( h^{-1}(x) \) are:
[tex]\[
\boxed{\text{Domain: } x < 4}
\][/tex]
[tex]\[
\boxed{\text{Range: } y < 5}
\][/tex]
