"C = Y^{2 } - 2'' is an equation, something that is either true or false when C and Y are given concrete numerical values. That's one thing that can be said for sure. It can beinterpretedas a function, but this requires several choices, and not all of them lead to a function.

The first choice is to select the dependent and independent variables. If the independent variable is Y and the dependent one is C, then this is indeed a function mapping every Y into Y^{2} - 2. If the independent variable is C and the dependent one is Y, then this is not a function because some values of the independent variable correspond to two different values of the dependent variable. E.g., C = 2 corresponds to Y = 2 and Y = -2. But even in this case this can be turned into a function by artificially restricting the codomain, i.e., the set where the dependent variable ranges. If we postulate that Y is nonnegative, then the mapping from C to Y becomes a function again (only for C >= -2).

A function is not only a law that provides some output for each input, but it also includes the domain and the codomain, the sets where the input and output come from. The same law may produce different functions for different domains, and it can even be or not be a function depending the domain and the codomain, as we have seen above.

There are other ways to interpret this equality. We can treat it as a family of constant functions from X to Y, two for each C > -2. For example, C = 2 gives functions Y(X) = 2 and Y(X) = -2.