https://file.aiquickdraw.com/mathbot/file/1726084300888nos9lfzc.jpg

The problem asks to find the inverse of the given function \( h(x) \) and to state the domain and range of \( h^{-1}(x) \), the inverse function, in interval notation.

The given function is:

\[
h(x) = \frac{7x - 3}{9x + 1}
\]

The inverse function has already been found:

\[
h^{-1}(x) = \frac{-3 - x}{9x - 7}
\]

We need to verify and provide the **range** of \( h^{-1}(x) \).

### Finding Domain and Range of the Inverse

#### 1. Domain of \( h^{-1}(x) \)
The domain of the inverse function corresponds to the range of the original function \( h(x) \).

For \( h(x) \), the vertical asymptote occurs when the denominator is zero:

\[
9x + 1 = 0 \implies x = -\frac{1}{9}
\]

Thus, the domain of \( h(x) \) is:

\[
\text{Domain of } h(x) = \left( -\infty, -\frac{1}{9} \right) \cup \left( -\frac{1}{9}, \infty \right)
\]

This implies that the range of \( h^{-1}(x) \) is:

\[
\text{Range of } h^{-1}(x) = \left( -\infty, -\frac{1}{9} \right) \cup \left( -\frac{1}{9}, \infty \right)
\]

#### 2. Domain of \( h^{-1}(x) \)

From the solution provided in the image, the domain of \( h^{-1}(x) \) is:

\[
\text{Domain of } h^{-1}(x) = \left( \frac{7}{9}, \infty \right)
\]

This corresponds to the range of \( h(x) \), which was restricted to values where the function \( h(x) \) is valid.

### Final Answer

- The **range** of \( h^{-1}(x) \) is: \(\left( -\infty, -\frac{1}{9} \right) \cup \left( -\frac{1}{9}, \infty \right)\)

Let me know if you want any further clarification.

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Here are 5 questions related to this concept:
1. How do we calculate the inverse of a rational function like \( h(x) \)?
2. What happens to the asymptotes of a function when finding its inverse?
3. How can we determine if a function has an inverse?
4. Why is it important to check the domain and range when working with inverses?
5. Can a function have more than one inverse?

**Tip**: Always check for vertical and horizontal asymptotes to understand the behavior of a rational function's domain and range!

Find the inverse of the function h(x) = \frac{7x-3}{9x+1} and state the domain and range of the inverse function in interval notation.

Learn how to find the inverse of the function h(x) = \frac{7x-3}{9x+1} and how to determine the domain and range of the inverse function. This example covers key concepts in rational functions, domain and range, and inverse function calculation.

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