A function g : S → T is said to be a left inverse for the function f : T → S if g◦f equals the identity function on $T$. In this case, f is also a right inverse for g . Prove explicitly that
That t any a function has a left inverse it is injective and if it has a right inverse it is surjective, give an example of a function that has 2 distinct left inverses.
I'm not too sure if i understand the difference the difference of between a left and right inverse to begin with. I think it's left inverse if in f: S --> T and g: T --> S meaning in f is you plug in S to f(x) you get T and for g(x) plugging in T you get S. So g(x) is the inverse of f(x) and vice versa.
Left inverse f(g(x)) = S and right inverse when rather g(f(x)) = T. Definitions have $1_S$ and $1_T$ which confuses me.
To be injective i also know that for every value of the codomain T there must be a unique value of S the domain. So there cannot be more than one value of T that maps into a value of S.
To surjective and everyone value of T the codomain must map to a value the domain S. So for example e^x is not surjective for all real x because its range is y > 0 rather than having a range of all real x. However you could cut the parameters to something like x>0 and even say it's bijective where it is both surjective and injective.