I was doing some exercises on discrete maths and came across this question I don't know how to solve, nor can I find any relevant examples in my book.
Suppose $f: \mathbb R \to \mathbb R$ where $f(x) = ⌊x/2⌋$, if $T = \left\{3, 4, 5\right\}$ , then what's the inverse of $f (T)$?
I know how to get $f (T)$ but how to get its inverse? Could anyone teach me please?
$\begin{array}{c} f & : & \Bbb R & \to & \Bbb R \\ & & x & \mapsto & \left\lfloor \dfrac x2 \right\rfloor \end{array}$
Before we start trying to solve equations, let's try to see a pattern. I looked for a few of the largest intervals that get sent into a few integers. \begin{align} [0, 2) &\to 0 \\ [2, 4) &\to 1 \\ [4, 6) &\to 2 \end{align}
It seems that, for $n \in \Bbb Z^+$, $f^{-1}(\{n\}) = [2n, 2n+2)$
Can you finish this?