To solve the given problem:

We are asked to find $f^{-1}(3)$, where $f^{-1}$ is the inverse of $f(x) = \frac{2x - 1}{3}$.

Steps:

1. Definition of the inverse: The inverse function $f^{-1}$ satisfies: $f(f^{-1}(y)) = y \quad \text{and} \quad f^{-1}(f(x)) = x$ Thus, we need to solve $f(x) = 3$ for $x$, which will give $f^{-1}(3)$.

2. Solve $f(x) = 3$: Given: $f(x) = \frac{2x - 1}{3}$ Set $f(x) = 3$: $\frac{2x - 1}{3} = 3$

3. Eliminate the fraction: Multiply through by 3: $2x - 1 = 9$

4. Solve for $x$: $2x = 10 \quad \implies \quad x = 5$

Thus, $f^{-1}(3) = 5$.

Final Answer:

$\boxed{5}$

Would you like a more detailed explanation, or do you have any questions?

Related Questions:

1. What are the steps to verify that two functions are inverses of each other?
2. How can you find the inverse of a linear function?
3. What is the domain and range of $f(x) = \frac{2x-1}{3}$?
4. If $f^{-1}(y) = 2y + 1$, how can we derive this expression algebraically?
5. How do you interpret the graphical relationship between $f(x)$ and $f^{-1}(x)$?

Tip: Always double-check your calculations when solving for inverses, especially with fractions.