Cool ways of defining functions.

[TylerH]

$$f(x)=x^3$$
or
$$f:\Re \rightarrow \Re$$
$$r \rightarrow r^3$$

What are some other ways to define functions? Exotic and extraneous as possible.

 

[JJacquelin]

I suppose that you now a cool way to define the function f(x) = ln(x)

 

[TylerH]

Yeah, that was my first example. :)

 

[HallsofIvy]

$$f(x)=x^3$$
or
$$f:\Re \rightarrow \Re$$
$$r \rightarrow r^3$$

What are some other ways to define functions? Exotic and extraneous as possible.

Actually, $f(x)= x^3$ does NOT define a function because it does not specify the domain. You might well assume that the real numbers is intended but why not the complex numbers.

As for the second form, it says too much. You don't have to specify that the range is the set of real numbers because if the domain is the real numbers and the "formula" is $x^3$, the range must be the set of real numbers.

If you want an exotic way of defining functions, how about this:
Bessel's function, of order 0 and of the "first kind" is defined as
"The function satisfying Bessel's equation of order 0,
$$x^2\frac{d^2y}{dx^2}+ x\frac{dy}{dx}+ x^2y= 0$$
and the initial conditions y(0)= 1, y'(0)= 0."

Or the "Lambert W function" which is defined as
"The inverse function to $f(x)= xe^x$".