Question

let A - (1;2;3) which of the following function f:A-A does not have an inverse function?

Answer 1

you didn't mention about options , I think question is --- let A={1,2,3}, which of the following functions f :A-A dose not have an inverse function -(a) f = {(1,1),(2,2),(3,3)}(b) f = {(1,2),(2,1),(3,1)}(c) f = {(1,3),(3,2),(2,1)}

solution :- Any function will be an inverse function, when function is one - one and onto.
check option (a) : f(1) = 1 f(2) = 2 f(3) = 3 , here it is clear that f has unique solution in it's domain. so, f is one-one function.
here also it is clear that, codomain = range = { 1, 2 , 3 } so, f is also onto function.so, f is inverse function. 
check option ( b) : f(1) = 2 f(2) = 1 f(3) = 1 here, f(2) = f(3) = 1 , hence, it is not one- one function . so, it is not inverse function.
check option ( c ) : f(1) = 3 f(3) = 2 f(2) = 1 , here f has unique value in each elements of f . so, f is one - one function.also codomain = range = { 1,2,3 } so, f is also onto Function.hence, f is inverse function.

so, option (b) doesn't have an inverse function.