Solve the equation y=\frac{2x+1}{x} for x in terms of y. Do this by first multiplying both sides by x to get xy=2x+1. Then subtract 2x from both sides to get xy-2x=1 and then factor to get x(y-2)=1. Finally, divide both sides by y-2 to get x=f^{-1}(y)=\frac{1}{y-2}.
Oftentimes people now "swap" the x and the y around to write the answer as y=f^{-1}(x)=\frac{1}{x-2}. There are two main reasons this is done: 1) people are used to using "x" for the independent variable and "y" for the dependent variable and, more significantly, 2) This swapping leads to the reflection property: the graphs of y=f(x) and y=f^{-1}(x) are reflections of each other across the 45^{\circ} line y=x.
As a purely symbolic matter, the swapping is not necessary. In fact, if the variables have specific real-life meanings, it's best not to swap them.
