Today I was thinking of discovering new functions. Many functions come from physics for example, where they are made to "model" some behavior.
But surely there are other sources for functions than physical things.
But how can one "gauge", whether such functions are useful or not? And for what they could be utilized?
I suspect that people don't usually invent new functions and see what they are good for, they try to solve problems that lead them to define a function. They write up the solution and others notice that the function helps with their problem, too. If a function shows up enough, we give it a name so we can talk about it. That may prompt others to work on the properties of the function. As you say, the trig functions show up lots of places. Once they show up so much we put a lot of work into understanding them.
One example would be solving the $14-15$ puzzle. Working on this naturally leads to the idea of even and odd permutations. If that idea wasn't already known you would define that function and use it in your solution. There are lots of other combinatoric problems that the parity of a permutation solves, so it would rapidly become popular.