Just as in case of f(x)=(x-2)^2+3, f is a function of x and when we try to draw such a function on say Cartesian coordinates, we use y=f(x). But x and y are just two variables and nature of function does not change, when we replace x by y and y by x.
However, a Cartesian graph of the function does change. This is as we always consider x as horizontal axis and y as vertical axis. We do not reverse these axes, but why we do not do that, because everybody understands that way and no body wants any confusion.
Similarly, in x=(y-2)^2+3 we have x as a function of y which can be written as x=f(y).
Further x=(y-2)^2+3 is an equation with two variables and hence we can express it both as x=f(y) as well as y=f(x). In fact solving for y we get y=sqrt(x-3)+2
However, there is a limitation as in x=f(y), we find there is an x for all values of y, but in y=f(x), y is not defined for x<3.