Why are the variables exchanged when finding an inverse function?

In mathematics, an inverse function is defined by interchanging the roles of the independent and dependent variables of a given function.1

The variables are exchanged because an inverse function is designed to "undo" the action of the original function, effectively reversing the relationship between the input and the output. In a standard function $y=f(x)$, the rule maps an input $x$ to a unique output $y$. By swapping these roles, the inverse function $g(y)=x$ maps the output back to the original input, returning the variable to its initial state.21

This process is a formal convention used to derive the inverse expression, such as when converting the function $f(x)=2x$ to its inverse $f^{-1}(x)=x/2$. By interchanging the variables, you align the new function's notation with the standard practice of expressing the dependent variable as a function of the independent variable $x$.1