The problem asks to find the inverse of the function $f$ which is defined as $f(x) = \frac{1}{3}(2x-1)$. We want to find $f^{-1}(x)$.

1. Problem Description

The problem asks to find the inverse of the function $f$ which is defined as $f(x) = \frac{1}{3}(2x-1)$. We want to find $f^{-1}(x)$.

2. Solution Steps

To find the inverse of the function $f(x) = \frac{1}{3}(2x-1)$, we can follow these steps:

Step 1: Replace $f(x)$ with $y$.

$$y = \frac{1}{3}(2x-1)$$

Step 2: Swap $x$ and $y$.

$$x = \frac{1}{3}(2y-1)$$

Step 3: Solve for $y$.

First, multiply both sides by 3:

$$3x = 2y - 1$$

Next, add 1 to both sides:

$$3x + 1 = 2y$$

Finally, divide both sides by 2:

$$y = \frac{3x+1}{2}$$

Step 4: Replace $y$ with $f^{-1}(x)$.

$$f^{-1}(x) = \frac{3x+1}{2}$$

3. Final Answer

$f^{-1}(x) = \frac{3x+1}{2}$