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The image you provided asks you to find the value of \( f^{-1}(a) \), given the following information:

### Function:
\[
f(x) = \frac{1}{3x}
\]
where \( a \) is a real number such that \( f(x) = a \).

### Problem Statement:
Find \( f^{-1}(a) \).

### Solution:

1. **Express the function equation:**
   \[
   f(x) = a \quad \text{which implies} \quad \frac{1}{3x} = a
   \]

2. **Solve for \( x \):**
   \[
   \frac{1}{3x} = a
   \]
   Multiply both sides by \( 3x \):
   \[
   1 = 3ax
   \]
   Now, solve for \( x \):
   \[
   x = \frac{1}{3a}
   \]

3. **Interpretation:**
   This \( x \) is the value of \( f^{-1}(a) \).

### Final Answer:
\[
f^{-1}(a) = \frac{1}{3a}
\]

Would you like further explanation or details? Here are five related questions that might be of interest:

1. How would you determine the domain of \( f^{-1}(x) \)?
2. What happens if \( a = 0 \)? Can \( f^{-1}(0) \) be found?
3. How can you verify that \( f(f^{-1}(x)) = x \) for this function?
4. What is the inverse function of \( f(x) = \frac{1}{2x} + 1 \)?
5. How does the graph of \( f(x) = \frac{1}{3x} \) compare to the graph of its inverse?

**Tip:** When finding the inverse of a function, switch \( x \) and \( y \) (where \( y = f(x) \)) and solve for the new \( y \). This new \( y \) represents \( f^{-1}(x) \).

Learn how to find the inverse function f^-1(a) for f(x) = 1/(3x) with a step-by-step solution. This problem involves understanding inverse functions and applying the inverse function formula. Suitable for advanced high school students.

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