I have several related questions, so I'm going to label them to make sure I understand what questions that answers are referring to. I understand that a function is an expression that produces one output for each set of inputs, but also,
a) could it be said that all functions are equations, but not all equations are functions?
In the fourth post on this page (http://www.physicsforums.com/showthread.php?t=449496) the writer takes the equation x^2 + y^2 + z^2 = 1 and rewrites it as z = f(x, y) = sqrt(x^2 + y^2) and then says “You can draw a line anywhere, so long as it's parallel to the z-axis and it will intersect the surface exactly once in the case of z=f(x,y). No so if f(x,y,z)=const.”
b) Is the writer saying that when f(x,y,z)=constant is written as z=f(x,y) it becomes a function?
If so, does this mean(?) that
c) z = f(x,y) = polynomial is always a function?
d) z = f(x,y) = polynomial = constant is always a non-function/equation?
e) Then if z = f(x,y) = polynomial = constant is always a non-function/equation, does this mean it is incorrect to say that the notation f(x,y) means “function of x and y”?
f) Equations have solutions/roots and functions have outputs?