elementary set theory - Build Bijection Function with Parameters

There are a,b $\in$ $\mathbb R$ such that a < b.
Show bijection function $f:[a,b) \to (0,1)$

I've tried something like:
$\\$ $f(n) =$ $$\begin{cases} n & \text{0 < $n$ < 1} \\ \frac{1}{n} & \text{$n$ > 1} \\ \frac{1}{2} & \text{$else$} \end{cases}$$

But that function isn't injective.

can I get any clue please ?