Find an example of sets X and Y a function fX Y and a subset

Find an example of sets X and Y, a function f:X Y and a subset S g X such that

Solution

Take the example of f(t) = cost

f(pi/4) = cos(pi/4) = 1/sqrt(2)

f(-pi/4) = cos(-pi/4) = 1/sqrt(2)

so, for t= pi/4 , f(t) = 1/sqrt(2)

f -1(1/sqrt(2)) can be either pi/4 or -pi/4

Hence F-1(F(s)) = s may not be always true, Hence showed the example