Question

let A={1,2,3}, which of the following functions f :A-A dose 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.