Question

Question about inverse function ?

Hi, I have some ? in my mind and I want to clear, someone answer, please:

a) A function which is two sided inverse (bijective) is inverse? ( two sided inverse = inverse? )

b) Every function have a inverse?

c) A function that is not injective or surjective have a inverse?

That's all, thanks for your answer

Answer 1

Please see below.

Explanation:

A function has an inverse function if and only if the function is injective.

(a) A function that has a two-sided inverse is invertible

`f(x) = x+2` in invertible. The inverse of `f` is `g` where `g(x) = x-2`.

I don't reacll see the expression "`f` is inverse".
In American English I see
"`f` is invertible (or invertable)"
and `f` has an inverse
and "`f` is the inverse of `g`"
But not "`f` is inverse"

(b) and (c) I think not.

Consider `f:ZZrarrRR` given by `f(n) = n^2`

`ZZ = {. . . ,-2,-1,0,1,2,. . . }`

It seems that this function has neither a left nor a right inverse.