inverse of composition of functions


Let f and g be invertible functions such that their compositionMathworldPlanetmathPlanetmath fg is well defined. Then fg is invertible and


Before proving this theorem, it should be noted that some students encounter this result long before they are introduced to formal proof. Fortunately, there is an intuitive way to think about this theorem: Think of the function g as putting on one’s socks and the function f as putting on one’s shoes. Then fg denotes the process of putting one one’s socks, then putting on one’s shoes. (Recall that function composition works from right to left.) Note that (fg)-1 refers to the reverse process of fg, which is taking off one’s shoes (which is f-1) followed by taking off one’s socks (which is g-1).

Due to the intuitive argument given above, the theorem is referred to as the socks and shoes rule. This name is a mnemonic device which reminds people that, in order to obtain the inversePlanetmathPlanetmathPlanetmath of a composition of functions, the original functions have to be undone in the opposite order.

Now for the formal proof.


Let A, B, and C be sets such that g:AB and f:BC. Then the following two equations must be shown to hold:

(g-1f-1)(fg)=idA (1)
(fg)(g-1f-1)=idC (2)

Note that idX denotes the identity function on the set X.

The two equations given above follow easily from the fact that function composition is associative.

(g-1f-1)(fg) =g-1((f-1f)g)
(fg)(g-1f-1) =f((gg-1)f-1)

The socks and shoes rule has a natural generalizationPlanetmathPlanetmath:


Let n be a positive integer and f1,,fn be invertible functions such that their composition f1fn is well defined. Then f1fn is invertible and


A sketch of a proof is as follows: Using inductionMathworldPlanetmath on n, the socks and shoes rule can be applied with f=f1fn-1 and g=fn.

Title inverse of composition of functions
Canonical name InverseOfCompositionOfFunctions
Date of creation 2013-03-22 17:47:47
Last modified on 2013-03-22 17:47:47
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 8
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 03-00
Classification msc 03E20
Classification msc 97D40
Synonym socks and shoes rule
Related topic Function
Related topic InverseFormingInProportionToGroupOperation