Recently, I have been thinking about functions that make functions based on a set of conditions. Originally, I thought this is what generating functions were but after doing some research, I didn't quite find what I was thinking about.
Specifically, I am talking about a function, $f$, (please mind the notation, I have no idea how to express this otherwise) that generates a "family" or "set" of functions given some conditions $k$.
So for example, if the conditions are that $f$ has to make a set of functions that is polynomial of degree $1$ $(ax + b)$ and has integer coefficients between $1$ and $10$ and the intercept is also made up of integer coefficients between $1$ and $10$ then that set of functions has $100$ polynomials in it.
In programming, I can make a pure function which does this. Below is the mathematica code.
(Range[1, 5] + #[x]) & /@ {a, b, c, d, e}
Flatten[%]
So, the above will generate
{1 + a[x], 2 + a[x], 3 + a[x], 4 + a[x], 5 + a[x], 1 + b[x], 2 + b[x],
3 + b[x], 4 + b[x], 5 + b[x], 1 + c[x], 2 + c[x], 3 + c[x],
4 + c[x], 5 + c[x], 1 + d[x], 2 + d[x], 3 + d[x], 4 + d[x], 5 + d[x],
1 + e[x], 2 + e[x], 3 + e[x], 4 + e[x], 5 + e[x]}
In this case, the conditions are that $y$-intercepts have to be between $1$ and $5$ and the $x$ can be any value {a, b, c, d, e}.
I was wondering what branch of mathematics this is related to and where can I find the necessary resources to learn more about this? Furthermore, does what I have above make sense?
Thanks a lot for your time!
P.S. Can someone tag this correctly?