I know that to graph $\arcsin x$, I need to only take into account its principal values, that is its image is restricted to $[-\frac{\pi}{2},\frac{\pi}{2}]$. Same with $\arccos x$, its image is restricted to $[0,\pi]$.
Now, say I want to graph a function like $f(x)=\frac{1}{2}\arcsin(4(x-1))+\frac{\pi}{2}$, what is the correct graph? Can I choose where it is restricted? When I use a graphing software it's restricted to $[\frac{\pi}{4},\frac{3\pi}{4}]$, but can I restrict the image to $[-\frac{\pi}{4},\frac{\pi}{4}]$ or any other interval that fulfills the definition of function?
No. The $\arccos$ and $\arcsin$ functions are defined by convention, just as you describe them in the first paragraph. If you plug the $\arcsin$ function with that definition into your $f(x)$, the image lies in $[\frac{\pi}{4},\frac{3\pi}{4}]$. That's not an arbitrary feature of your graphics software; it's a necessarily result of the conventional definition. It wouldn't make much sense to define $\arcsin$ differently in different contexts.