I just learned about an area of math called functional analysis. Have you ever heard of it? It is sort of like algebra, but works with statements about functions to solve for them rather than working with statements about numbers to solve for them.

Here is a very simple example of a problem in functional analysis:
Find an expression for f(x) such that f(x+3) = x^2 + 2x
If you get stuck, think of how you would solve for f(7)

From what I have read, functional analysis has lots of practical applications in modeling the world, much the way that differential equations does. I am surprised that, even though I have a degree in math, I never heard of this before.