implicit function f(x,y) and z=f(x,y)

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Joined Apr 15, 2011

In my mind I'm confusing an implicit function f(x,y) with function z=f(x,y). One of the characteristics of an implicit is that it can let us represent a two-part function in one expression. For instance, consider the equation of a circle: x^2 + y^2 = c^2 [one part function] => y=+/- sqrt(c^2-x^2) [two part function]. Are there some other advantages of an implicit function? In case of an implicit function, y is a dependent variable (function of x) and x is independent.

Isn't function in space also represented as z=f(x,y) where 'z' is determined using x and y coordinates? Is y an independent variable or function of x as it is for an implicit function in x-y plane? Perhaps, both x and y are functions of some other parameter such as 'c'.

I'm sorry if my query(ies) is too confusing. Please let me know if you need some clarification about any part. Please help me with it. Thank you.



Joined Jul 7, 2009
An implicit function is just a more general way of writing functions. Beginners often think that everything should be written like y = f(x) or z = g(x, y), but many functions cannot be solved for the individual variables. When you study advanced calculus, you'll learn about the important implicit function theorem, which gives conditions under which you can (locally) solve for the individual variables.

So, to answer your first question, implicit functions let you write down equations that can't be written in the y = f(x) form.

Similarly, in more dimensions, the more general form is to write the functional relationship implicitly. Thus, for example, a relationship amongst the three Cartesian coordinates x, y, z might be written as h(x, y, z) = 0. You might or might not be able to write it as z = F(x, y).

Now that you know this, this can answer one of your other recent questions about what's special about finding the roots that make an equation zero. It's because we often 1) have a relationship amongst some variables and 2) we know the values of all of them except one and want the value of that last one. In many real-world problems, part 2 can involve needing to find a root. If the equation is given implicitly, now you can see why you need to find roots -- because you can't solve algebraically for the variable wanted. Thus, you'll have an equation like f(x, y, z) = 0 where you know x and y (say, as 2 and 3.5) and you want to know z. Then, in general, you have to solve for the root of the equation f(2, 3.5, z) = 0.

FYI, also make sure you learn the details about using parametric forms of the equations you're dealing with. You can do things in practical math with parameters that you can't do with the explicit or implicit functions. As an example, I recently solved the problem of two pipes intersecting each other and making a template from a piece of paper to cut the shape of the intersection on one of the pipes. It turns out it's easy to do with freshman-level analytic geometry and parametric equations. In fact, the equations easily generalize to handling any arbitrary cylindrical shapes (i.e., any shape of cross section), but in this case you can't do the algebra explicitly because there's an implicit equation that can't in general be solved. But, for a practical problem, you'd write a root-finder in code and get the answer that way.