The following function is one-to-one. Find the inverse of the function and graph the function and its inverse on the same set of axes. \[ f(x)=2 x-1 \] \[ f^{-1}(x)= \] (Type a simplified fraction.)

Step 1 ：The given function is \(f(x)=2x-1\).

Step 2 ：To find the inverse of a function, we swap the x and y (or f(x)) values. This means we replace every x in the original function with y and solve for y.

Step 3 ：Replacing x with y in the original function, we get \(f(y)=2y-1\).

Step 4 ：Solving this equation for x, we get \(x=\frac{y}{2}+\frac{1}{2}\).

Step 5 ：So, the inverse of the function \(f(x)=2x-1\) is \(f^{-1}(x)=\frac{x}{2}+\frac{1}{2}\).

Step 6 ：\(\boxed{f^{-1}(x)=\frac{x}{2}+\frac{1}{2}}\) is the final answer.