Is the equation of a circle a function? We know, for a circle of radius 1 centred at the origin: $$ y^2 + x^2 = 1$$
Now when I saw the graph of the given equation- any x-value in within the domain has two images, but how can that be? 
Since I've read that in a function any pre-image cannot have two images.

And now the above can be equivalently written as, $$y = \sqrt{1-x^2} $$ But now, surprisingly(for me), any x-value doesn't have two images!
But why is that?
Because there is also $$y=-\sqrt{1-x^2}.$$

The circle with equation $x^2+y^2=1$ it's not a graph  of a function.

We can understand it by another algebraic way.
$$\left(\frac{3}{5},\frac{4}{5}\right)$$ and
$$\left(\frac{3}{5},-\frac{4}{5}\right)$$ 
placed on the graph and this is a contradiction with a definition of the function.
Something is a function if, and only if, there is only one value of y for each value of x. That's the definition of function.

Why is so? Because the idea of function is that you can relate two values in a determinist way, so that you know the value of the dependent variable if you are given the value of the independent one. If however there are two 
(or more) different possible values for y for a given value of x, then y doesn't actually depend exclusively on x, ie, it isn't a "function of x".