Let fxx3 We learned that the floor and the ceiling functions

Let f(x)=?x/3?. We learned that the floor and the ceiling functions are NOT invertible, but we also learned about the set of preimages of any value in the Range, the set of images. Keeping that in mind, give your answer in interval notation if necessary.

(a) Find f?1({3}).

Solution

(a)
So, you are trying to find ALL points which map to 3 via f. Well all the numbers that ceil to 3 are:
2 < x <= 3, and so hence the numbers which are taken to 3 via f are in this range multiplied by three:
f^-1({3}) = (6, 9]

Answer ---> (6 , 9]