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The given function is \( h(x) = \frac{3x + 1}{9} \), and the question asks for its inverse, if it exists.

To find the inverse of \( h(x) \), follow these steps:

### Step 1: Replace \( h(x) \) with \( y \)
\[
y = \frac{3x + 1}{9}
\]

### Step 2: Swap \( x \) and \( y \) (this is the step that begins finding the inverse)
\[
x = \frac{3y + 1}{9}
\]

### Step 3: Solve for \( y \)

- Multiply both sides by 9 to eliminate the denominator:
\[
9x = 3y + 1
\]
- Subtract 1 from both sides:
\[
9x - 1 = 3y
\]
- Divide both sides by 3:
\[
y = \frac{9x - 1}{3}
\]

### Step 4: Rewrite the inverse function
\[
h^{-1}(x) = \frac{9x - 1}{3}
\]

Thus, the inverse of \( h(x) = \frac{3x + 1}{9} \) is:
\[
h^{-1}(x) = \frac{9x - 1}{3}
\]

Would you like a more detailed explanation of any step? Here are some related questions to extend your understanding:

1. How can you verify that \( h^{-1}(x) \) is the correct inverse?
2. What are the conditions under which a function has an inverse?
3. Can all linear functions be inverted?
4. How does the domain and range of a function change when considering its inverse?
5. How can graphing both \( h(x) \) and \( h^{-1}(x) \) help visually confirm the inverse?

**Tip:** To find the inverse of a function, always swap the roles of \( x \) and \( y \), then solve for \( y \).

Which of the following is the inverse of h(x) = (3x + 1)/9, if it exists?

This guide explains how to find the inverse of the function h(x) = (3x + 1)/9 through a detailed step-by-step algebraic process. Understand key concepts such as inverse functions and apply algebraic techniques to solve for the inverse.

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