What do we call a law that takes a number as input and returns a function? A law that turns a number to a number is called a function. A law that turns a function into a number is called a functional e.i. y=max(f(x)).
We call a law that converts a function into a function (e.i. a derivative) an operator.
What do we call a law that converts a number into a function?
 A: I would call something that maps functions to functions an operator. Differential operators are a class that is often studied.
The term function can be understood more generally. Essentially anything that maps from one space to another as long as each element from the first space is mapped to exactly one element of the second is a function. Either space could be just natural/ real/ etc numbers but could also be a space of functions or a space of matrices or something like that.
Edit in response to comment: A space of functions is usually much much bigger than a space of numbers. If you have a function that maps the real numbers to functions from real to real numbers only a tiny amount of the functions will be in the image of your mapping so this kind of things is rarely useful.
The only context where one would use something similar is if you have a family of functions parametrized by numbers. For example the family of functions $f_n(x)=x^n$ where $n$ is a natural number but I wouldn't really think of this as a mapping from numbers to functions.
A: We could call this law that converts a number into a function a representation. The representation is well-defined, if we restrict ourselfs to some subset of functions.
For example, given a number $k \in \mathbb{R}$ we can define a function $f(x) = k x$ for all $x \in \mathbb{R}$ that represents the number $k$. In this trivial example, for the set of functions $\lbrace f(x) = kx$ for all $x \in \mathbb{R} \mid k \in \mathbb{R}\rbrace$, the inverse $f \mapsto k$ is bijective, i.e. there exists a unique $k \in \mathbb{R}$, such that $f(x) = kx$ for all $x \in \mathbb{R}$. This is also a corollary of Riesz representation theorem.
