Let X: [0,1] being in all real numbers be given by X(x) = max{x-0.5,0}. Find the inverse images of all intervals [a,b]

Question

Let X: [0,1] being in all real numbers be given by X(x) = max{x-0.5,0}. Find the inverse images of all intervals [a,b]

Due to lack of examples related to the theory, I am unable to properly write the solution step by step, though I know intuitively the answer.

Could anyone here be willing to provide a solution to the question above

Answer 1

The inverse image of an interval [a,b], X^-1([a,b]), is the set of all points x in [0,1] such that a <= X(x) <= b. Note X(x) >= 0; if b is negative, it follows that X^-1([a,b]) = ∅. If b is positive, then X^-1([a,b]) is [a + 0.5, min{b + 0.5, 1}] for a > 0 and [0, min{b + 0.5, 1}] for a <= 0. Indeed, by definition of X, X(x) = 0 for 0 <= x <= 0.5 and X(x) = x - 0.5 for 0.5 <= x <= 1. So for 0 < a < b, a <= X(x) <= b iff a <= x - 0.5 <= b iff a + 0.5 <= x <= min{b + 0.5, 1}. For a <= 0 < b, a <= X(x) <= b iff 0 <= x  and x - 0.5 <= b iff 0 <= x <= min{b + 0.5, 1}.