Implicit Functions | Click here to open an associated Mathcad worksheet: |

We're used to seeing algebraic definitions of functions in which the output y is given **explicitly** in terms of some combination of the input x :

We think of these rules as accepting the input x , going through a series of steps that transform x , and then announcing the final result as y . The output y is not involved in the "business" of computing the function, it is only the end product.

What, then, do we make of equations like this:

Here the x s and y s are all mixed-up together, on either side of the equation. There is no obvious way to proceed from an "input" x to an "output" y . It isn't even clear if each y is actually a function of the corresponding x .

We could, of course, attempt to solve each of these equations for y , writing y explicitly as a function of x , and so try to put ourselves back on familiar ground. In all three cases, however, we will fail.

Sometimes the reason we will fail is that y is not uniquely determined by the value of x : *many* y s will balance the equation when we plug in a particular x. (For example, solving the first equation for y leads to .) Sometimes the reason is a failure of algebra itself: we *can not* solve for y ; it is just too mixed-up with the x s for us to be able to extract it. (Try, for example, to solve the last equation for y .)

The y s in such "mixed-up" equations are often called **implicit** functions of x . Using the word "function" here is tricky, because, as we have seen, there may in fact be many y s corresponding to a particular x . What is "implicit" (the dictionary defines this as "implied or understood although not directly expressed") is the understanding that *many* functions may be lurking in such equations, if only we could coax them out.