On which of these domains is $f(x) = x^{2} - 4x$ one-one?

Explanation

1. 01

   $f(a) = f(b)$ gives $(a - b)(a + b - 4) = 0$, so a repeat needs $a + b = 4$ with $a \ne b$: two points on either side of 2.

2. 02

   On $[2, \infty)$, $a + b = 4$ forces $a = b = 2$: one-one. Each other domain holds such a pair: $f(1) = f(3) = -3$, $f(0) = f(4) = 0$.