Given two functions f and g, what is the inverse of g @ f ?

1 Answer
Dec 18, 2017

Assuming f and g are both invertible, you can say that (g circ f)^{-1}=f^{-1} circ g^{-1}. This is sometimes called the "socks-shoes property".

Explanation:

This can be verified using the associative property for function composition:

(g circ f) circ (f^{-1} circ g^{-1})=g circ (f circ f^{-1}) circ g^{-1}

=g circ id circ g^{-1} = g circ g^{-1} = id,

where id represents the "identity function" id(x)=x for all x.

The property is called the "socks-shoes property" because if f represents "putting your socks on" and g represents "putting your shoes on", then g circ f represents "putting your socks and shoes on" and (g circ f)^{-1} represents "taking your socks and shoes off". The fact that (g circ f)^{-1}=f^{-1} circ g^{-1} means that, to do this, you must "take your shoes off" (apply g^{-1}) before you "take your socks off" (apply f^{-1}).

A similar type of thing happens with matrix multiplication, or indeed, in any group G. Given a group G and a,b\in G, (ab)^{-1}=b^{-1}a^{-1}.