Function

A function \(f : A \to B\) requires two things: every element of \(A\) has an image, and that image is unique in \(B\). Depending on further conditions, it is injective if different inputs give different outputs, surjective if every element of \(B\) is reached, and bijective if both hold.

Take \(f(x) = x + 1\) from \(\mathbb{R}\) to \(\mathbb{R}\). It is injective because \(x + 1 = y + 1\) implies \(x = y\), and it is surjective because for any \(y\) the choice \(x = y - 1\) works. Hence it is bijective and its inverse is \(f^{-1}(y) = y - 1\).

A common mistake is to think that \(f(x) = x^2\) from \(\mathbb{R}\) to \(\mathbb{R}\) is injective just because it gives one value for each \(x\). Seeing that \(f(1) = f(-1) = 1\) with \(1 \neq -1\) rules this out. Also do not confuse codomain with image: the codomain is \(B\), while the image can be a proper subset of \(B\).