Editing Inverse function (section)

====Two-sided inverses====
An inverse that is both a left and right inverse (a '''two-sided inverse'''), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called '''the inverse'''.

: If <math>g</math> is a left inverse and <math>h</math> a right inverse of <math>f</math>, for all <math>y \in Y</math>, <math>g(y) = g(f(h(y)) = h(y)</math>.

A function has a two-sided inverse if and only if it is bijective.
: A bijective function {{mvar|f}} is injective, so it has a left inverse (if {{mvar|f}} is the empty function, <math>f \colon \varnothing \to \varnothing</math> is its own left inverse). {{mvar|f}} is surjective, so it has a right inverse. By the above, the left and right inverse are the same.
: If {{mvar|f}} has a two-sided inverse {{mvar|g}}, then {{mvar|g}} is a left inverse and right inverse of {{mvar|f}}, so {{mvar|f}} is injective and surjective.