Find the function to show two intervals are equivalent

Question

To show that two intervals are equivalent sets, you must show there is a bijection from one set to the other. My issue, is how do you determine the function? For example, say I would like to show that $[4,5)$ and $(-1,1]$ are equivalent, since these are two half open intervals, finding a function that satisfies this seems challenging.

Answer 1

Lets just talk about how one might go about this. Obviously, the points $5$ and $-1$ are equivalent and the points $4$ and $1$ are equivalent. So, what sort of function can send $f(5)=-1$ and $f(4)=1$? Yes, we actually don't require $f(x)$ to be defined for $x=5$ but there is no harm in using this as a starting point. But alright, we basically have two intervals so we might as well link them by a linear function

Set $f(x)=ax+b$ with the conditions stated above. Then solving gives $f(x)=-2x+9$ as the bijection between the intervals.