Does anyone else think should be the maximum of ?
You have . You know that g is one-to-one.
So the question is really: Given the restriction of the domain of f to [1, a], what is the largest value of a such that f(x) is one-to-one.
If your then it looks like x = 1 is on the left of the vertex, so a would have to be the x coordinate of the vertex of f(x). (That's the accurate way to say what I think you were trying to say.)