What are the requisites to define a function? I was reading on the Lambert W function, how it is the inverse relation of $xe^x$ and I noticed that there is no actual expression in terms of other functions for this relation. It seems like they just tacked a $W$ on to the argument and called it a new function, this is obviously not the case. However, it got me wondering how is it possible to define a new function in a useful way like the Lambert W function? Is it possible, for example, to define the inverse relation of $x+\sin x $? What about more complicated expressions like $x(\ln(x\ln x))$? My main question is: what are the requisites for this definitions to be useful and not mere symbols written on a page?
 A: Any function $f: X \rightarrow Y$ which is bijective (that is, the range of $f$ is $Y$, and also $x \neq y \Rightarrow f(x) \neq f(y)$) admits an inverse function $g: Y \rightarrow X$.  By definition, an inverse of $f$ is a function $g$ such that $g(f(x)) = x$ for any $x \in X$, and $f(g(y)) = y$ for any $y \in Y$.  There may or may not be a nice closed formula for $g$ (even if there is one for $f$), but whenever $f$ is bijective, it nevertheless exists.  
When $f$ is not injective (for example, the exponential function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(x) = e^x$, or the Lambert W-function), there isn't really an inverse function, but there is a notion of a "multivalued function" (which is not, strictly speaking, a function; a function needs to have precisely one output at each input).  See https://en.wikipedia.org/wiki/Multivalued_function
"Useful" functions are functions that you end up using when you solve problems.  Some functions, like the exponential function or the function $f(x) = xe^x$, are used in solving so many problems, that it becomes desirable to deeply study the functions themselves.  We may even be interested in describing the properties of the "inverses" of these functions (even though they aren't true inverses, since $e^x$ and $xe^x$ are not bijections).  It's all fairly subjective.  
However, just because you are studying the properties of some function, doesn't mean you can find a nice expression for that function (that is, an expression in terms of familiar "elementary functions" $x, e^x, \cos, \sin$ etc.).  There is no reason to believe any given function you encounter should be able to be described in a nice way like that.  For example, the antiderivative of $e^{x^2}$ shows up in probability (normal distribution), but that function cannot be described in any elementary way (well you can use an infinite series).
EDIT: I think I severely misunderstood your question.  When you said, "Define a function," I didn't realize you meant giving a function a formal name.  I think it all comes down to convention.  If a function gets used a lot, it gets named so people don't have to keep writing out the formula or saying "The inverse of such and such" etc.
