Comparing and contrasting equations and functions 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?
Thank you!
 A: If a mapping assigns a unique element in the range for each element in the domain, then this mapping is a function.
a. In 2-dimensions if the graph passes the vertical line test, then for each $x$ in the domain there is only one element $y$ in the range (since the vertical line only crossed the graph at one point) and hence this is a function.
In 3 dimensions, the domain can be considered to be the set of all pairs of $x$ and $y$.
b. Once you write it as $z = f(x,y)$, then for each $x$ and $y$ you have a unique $z$ and hence is a function. Note that there is $z$ term on the right hand side.
c. $x = f(x,y) = $ polynomial is a function (see b).
d. $z = $ constant is a function (see b)
e. Yes, $z = f(x)$ is read as $z$ is a function of $x$.
f. A common way of thinking of a function $f(x)$ is that it takes an input $x$ and maps it to $f(x)$, the output.   Solutions to equations are those values that make the equation true. So for $x + 3 = 0$, the solution is $x=-3$.  It does not make sense to talk about a solution to a function, $f(x)$, but one can talk about, for instance, a solution to $f(x)=0$, which are the zeros of the function.
Note:

*

*The equation $x^2 + y^2 + z^2 =1$ is not the same as $z = \sqrt{1-x^2-y^2}$, as the latter is only the top portion of the sphere (since the use of the square root implies the principal square root only, ie, $z \ge 0$).

A: A function associates every ( except where it is undefined) points in co-ordinate space with a certain unique value. It can associate either a vector or a scalar. When we write $f(x,y,z)$=constant we mean the points satisfying this relation form a surface in space. There is no sense of association of points with values. But if you could modify same relation into $z=f(x,y)$ for every $(x,y)$ you could associate a value to $z$. Thus you could say $z$ is the function of $x$ & $y$.
Yes $z=f(x,y)$ is a function
NO, $z=f(x,y)$=constant is also a function called constant function
what makes you think that it is incorrect to say $f(x)$ is a function of $x$?
Ya you can say equations have roots (points satisfying them) and functions have outputs
