I don't really understand what you mean. You can never divide by zero. You can't treat the functions as the same because they are not the same function. It's not a fundamental premise of algebra that you should be able to.
The corner cases aren't nasty, you have to take care of them every time. Like with your example, the solution is f(x)=x+2, except where x=-2 where f(-2) can be any number. There are a continuum of solutions to the equation each with a different value of f at x=-2.
The corner cases aren't nasty, you have to take care of them every time. Like with your example, the solution is f(x)=x+2, except where x=-2 where f(-2) can be any number. There are a continuum of solutions to the equation each with a different value of f at x=-2.