Prompted by your reply, I looked up 1-1 functions and learned its just a function that passes both the vertical and horizontal line test and that there is a test to check if a function is 1-1. With a quick observation, it doesn't look like the function x = t^2 +1 is 1-1, but according to your explanation, the function must be 1-1 if its independent variables can be isolated to one side. I just read that a function can be determined to be 1-1, thus invertible, by proving f(a) = f(b), however, this doesn't seem to be provable on the function x = t^2 +t (or at least I'm having trouble with it). If not provable, then its not 1-1, then not invertible, yet we still were able to solve it for t (get t to one side).
Either f(a) = f(b) is actually provable and is in fact 1-1, which is why its independent variable t can be isolated to one side. Or f(a) = f(b) is not provable and not 1-1, yet can be isolated to one side regardless (which would make less sense unless there's an additional fact about isolating the variable to one side that I still don't know about).