# Surjective functions

1. Mar 10, 2011

### SD123

1. The problem statement, all variables and given/known data
i) Show explicitly that any non-injective function with a right inverse has another right inverse
ii) Give an example of a function which has (at least) two distinct left inverses

2. Relevant equations
i) I don't believe there are any.
ii) "

3. The attempt at a solution
i) Since it says "explicitly" I doubt that my attempt would count as a solution even if it were correct, but here it is;

Since the function is non-injective and has a right inverse, it must therefore be surjective as it cannot be bijective, and in order for this function to be surjective it must have at least one more right inverse.

I also tried showing first that f(x) = x^3 - x is non-injective and then showing that the values x = -1,0,1 map to f(x) = 0 that I would be showing this is true but I am not sure if it is correct.

ii) For this I am really not sure if there is an easier way to do it apart from trial and error but I understand the theory behind it.

SD123

2. Mar 10, 2011

### HallsofIvy

Staff Emeritus
Try something as simple as this: f:{a, b, c}--> {x, y} defined by f(a)= x,f(b)= x, f(c)= y. g:{x, y}-->{a, b, c}, defined by g(x)= a, g(y)= c is a right inverse.

Think about that example to prove (i). However, note that an example is NOT a general proof.