# How do you prove that an implicitly defined function cannot be made explicit?

Consider for example the function:

$$x^2 e^y + \log(x)y^2 = 0$$

I suspect that neither x nor y can be isolated in this function (that it, it cannot be written as either $x(y)$ or $y(x)$. However, I can't really prove that other than to say "it's difficult to do if not impossible".

Is there a method by which it can be proven that no finite number of algebraic operations will lead to such a simplification? (The above function was presented as an example, but I am looking for a general solution)

• i think that is impossible – Dr. Sonnhard Graubner Apr 10 '17 at 18:25
• I think it depends on what elementary operations you allow. Your original expression contains $log$ so apparently at least some "common" functions are allowed in your answer. Specifically, do you consider the Lambert W function permissible in the solution? In which case $y=2W(\frac{-\sqrt{log(x)}}{2x})$ is a solution. – Χpẘ Apr 10 '17 at 18:28
• @Dr.SonnhardGraubner: Is it that you think the general method does not exist, or that it is not possible to isolate for any of the variables? – Michael Stachowsky Apr 10 '17 at 18:30
• @user2460798: Overall, I would like to prove whether or not a method exists to write the function as either $x = f(y)$ or $y = f(x)$. The functions required to do so may be horrifying, but that's OK. – Michael Stachowsky Apr 10 '17 at 18:31
• I'd suggest looking at math.stackexchange.com/questions/43263/…, which doesn't directly address your question, but is relevant. My sense is to copy how explicit defined functions are usually presented: using "well known" functions, certainly, log, exp, trig, arctrig, hyperbolic trig, hyperbolic argtrig, etc. Power series also make sense. Beyond that I'm inclined to believe it depends on your audience, and tradition of the math discipline you're targeting - such as well known integrals, special functions, etc. – Χpẘ Apr 10 '17 at 18:53

Considering some general implicit definition of the form $f(x,y) = 0$, this cannot be written as $x(y)$ if there exists a $y$ for which there are two $x_1, x_2$ with $f(x_1,y) = f(x_2,y) = 0$ and $x_1 \ne x_2$ as this would imply $x(y) = x_1$ and $x(y) = x_2$ in contradiction to $x_1 \ne x_2$, so $x(y)$ could not be a function. Uniqueness of the solutions of $f(x,y) = 0$ for any $y$ would therefore be necessary for $x(y)$ to be a function.
• I don't think the OP meant a function in the narrow sense - in the same way that the Lambert W function is not a function when argument $1/e<x<0$. I think he meant a relation specified by an implicit formula. – Χpẘ Apr 10 '17 at 19:35