Does this mean we need to find a function f such that any values of x,y,z will always yield 0? If so, how is the equation in next slide always yielding 0? If I put x = 1, I get f = 1-1.23 = 0.23 != 0..... :/
What is means is that if f(your_point) = 0, then your_point satisfies the function (i.e., your_point is on the line or plane or complicated implicit surface defined by this equation). If f(your_point) != 0, then your_point does not satisfy the function and it is not contained in the shape that you have implicitly defined.