Describing a function with independent variables.

Question

The normal distribution is often written $f(x\,\rvert\,\mu,\sigma^2)$, with the independent variables on the right side of the vertical line. Is this notation common, or are alternatives? For instance, is there a way of splitting the function in two, one with the independent variables and one with the dependent ones? e.g. $\phi(\mu,\sigma^2)\circ \varphi(x)$, or is it simply too common to just assume that all variables not explicitly mentioned as a function parameter are independent variables?