Functions

I've been thinking about a way to manually limit the domain of a function without writing next to the function the domain, for example dealing with real number the function $f(x)=arcsin(x)$ is limited by $-\frac{pi}{2} <= x <= \frac{pi}{2}$

I was thinking about a modulus graph and how the graph $f(x)=abs(2-x) + abs(2+x)$ is flat between x=-2 and x=2

if you were to line this flat bit up with y = 1, and then reflect the function you could have that for x<-2 y<0 and x>2 y<0.

so if we were to square root this function we could have a multiplier that between values we could specify the multiplier does nothing, but outside this it turns the function complex.

this is only approximate because the liner line we put take the modulus of in $sqrt(21-(abs(10-x) + abs(10+x)))$ would have to have an infinite gradient for this to work exactly, I'm curious if there are any better way to do this, using my multiplier we get graphs like this limited between x=-21/2 and x=21/2

https://twitter.com/fluffehadam/status/521708820108959744