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

1 Answer
Jan 23, 2018

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.