Yeah, so the problem is that the units of a function applied to a value is not necessarily the same as the value. Consider the area of a square. It has one argu

Yeah, so the problem is that the units of a function applied to a value is not necessarily the same as the value. Consider the area of a square. It has one argument whose unit is of length type. But, the function returns something of area type - totally fine. If you want to talk about e^(x) then you're totally free to do that but, you're right: if x is not dimensionless then the expression has no physical meaning (as little meaning as one meter added to one second, as an example of another clearly nonsensical physical function). In Physics, your starting point always makes sense dimensionally (F=m*a, E=m*g*h) and the game is to combine expressions that make dimensional (and physical) sense to glean insight into how different physical constraints of the system are related. If you have any Physics problem where you end up with e^(x) or sin(x) and x is not dimensionless then it's a dead giveaway that you performed an algebraic misstep at some point when playing the game. So, in short, you can propose a lot of functions that make no sense from a physical standpoint, but you won't arrive at those if you're working from first principles which are, by nature, dimensionally consistent.