How are you over-thinking it? I'll give you some hints. In general, it's better to try a simple solution for each of these than a complicated one.
For a) think really simple.
For b) write down the integers from -3 to 3 and the integers from 2 to 8 and try to get a function from the first to the second in an easy, systematic way by drawing arrows.
For c) you can think about functions that go down and then up.
For d) you can think about functions that go up and down but seems to bounce between two values (then you can scale and translate to get them between a and b)
For e) maybe you can think about functions that go bad at 2 and functions that don't exist for x<2, and maybe combine them.