The difference between inverse function and a function that is invertible?

Question

My discrete mathematics book says:

But I read an answer, https://math.stackexchange.com/a/2415543/390226, said:

["...]There are invertible functions which are not bijective,[..."]

And to the same question in the link, an answer said:

["]A function is invertible if and only if it is injective[."]

So for a function to have a inverse, it must be bijective. But any function that is injective is invertible, as long as such inverse defined on a subset of the codomain of original one, i.e. the image of the original function?

Answer 1

It all depends on the co-domain of your function.

When you have a function $$f:A\to B$$ which is one-to-one but not onto $B$, you may restrict your co-domain to a subset of $B'\subset B$ which is the range of $f$.

For example $$f:N \to N $$ defined by $$f(n)=2n$$ is not onto but it is one-to-one.

If we define, $$f^* : N\to 2N$$ with the same definition $f^*(n)=2n$

We have an inverse function, $(f^*)^{-1} (n) = n/2.$

Answer 2

When people said a function is "invertible", they mean it can be made invertible. And the rigorous definition of inverse function of $f$ in my book is:

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