R = {(a,a),(a,b)}. find the inverse of R and if the inverse of R is a function. why or why not?

Answer 1

R itself is not a function, because it sends the input "a" to two different outputs: "a" and "b".

To find its inverse, flip each ordered pair around:
R' = {(a,a),(b,a)}.

This IS a function, because each input is sent to only one output. On the input "a", it's unambiguous that the output should be "a". On input "b", the output is, again, unambiguously "a".