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Prove that $f:[0,2)\to [5\color{red},\color{\black}6)\cup [7,8)$What to do next?
Prove that $f:[0,2)\to [5.6)\cup [7,8)$
$$f(x)=\left \{ \begin{matrix} x+5& \mbox{ if }& x\in [0,1)\\x+6 & \mbox{ if }&x\in [1.2)\end{matrix}\right.$$
is injective and surjective.
Of course, my fingers were clumsy.Did you mean [5,6) and not [5.6)?
Also, [1,2) and not [1.2)?