How to use function notations when the function isn't straight forward

x² + y² = 4 is *not* a function, at least not in x. That's why you can't write it in function notation. A function must have one and only one value for each input, but because you can have two values for one x (e.g. 2 or -2 for x = 0), it's not a function.

You have two options.

You can go with an implicit definition like you mentioned, which is where you take two inputs and set it equal to a number. E.g. R(x,y) = 4, where R(x,y) = x² + y², but that's not probably that helpful here as it's so similar to the original equation.

Or you can go with a union of functions in x, like fermat9990 mentioned.

x² + y² =4 is the union of two functions:

fx)=√(4-x²) and g(x)=-√(4-x²)

noting down functions implicitly is a way to define a function. But you need to show it is actually a function. In your circle example y(x) is not a well defined function since for most x there are 2 values y satisfying the implicit equation.

Also those have a rather simple explicit form using square roots. But there are indeed many functions which can only be defined implicitly.

Another method is called parameterization in which one chooses a third variable, call it t, and breaks down x and y as functions of t. For example, for a circle we can parameterize through its angle measure.

Take x(t) = cos(t) and y(t) = sin(t) then clearly x²(t) + y²(t) = 1, regardless of the value of t, Pythagorean Trig Identity. And so, if we know x then we can use cos^-1(x) = t then plug that into y(t). I.e. y = sin(cos^-1(t)).

For a more general case we would use x(t) = r•cos(t) and y(t) = r•sin(t) for a fixed radius r > 0. Resulting in t = cos^-1(x/r) and y = r•sin(cos^-1(x/r)). For your problem take r = 2.

you would have to derive the equation to get the y value by itself. Either way, a circle is one of the four conic sections. You graph it geometrically, it really shouldn't be graphed as a function. Hope this helps!