Composition of Functions

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 21 September 2009

Definitions

Let $f:S\to T$ and $g:T\to U$ be functions. The composition of $f$ and $g$ is the function $g\circ f$ given by $$\begin{array}{rcl}g\circ f:S& ⟶& U\\ s& ⟼& g\left(f\left(s\right)\right)\text{.}\end{array}$$

Let $S$ be a set. The identity map on $S$ is the map given by $$\begin{array}{rcl}{\iota }_S:S& ⟶& S\\ s& ⟼& s\text{.}\end{array}$$

Let $f:S\to T$ be a function. The inverse function to $f$ is a function $f^{-1}:T\to S$ such that $$\begin{array}{rcl}f\circ f^{-1}& =& {\iota }_{\mathrm{T}}\text{,}\text{and}\\ f^{-1}\circ f& =& {\iota }_{\mathrm{S}}\text{,}\end{array}$$ where ${\iota }_T$ and ${\iota }_S$ are the identity functions on $T$ and $S$ respectively.

Visualisation [NOT SURE ABOUT THIS TITLE]

If we visualise functions as graphs, the identity function ${\iota }_S$ looks something like the image below.

In the pictures below, if the left graph is a pictorial representation of a function $f:S\to T$ then the inverse function to $f$, $f^{-1}:T\to S$, is represented by the graph on the right; the graph for $f^{-1}$ is the mirror-image of the graph for $f$.

Pictorially, graph (d) below, represents a function $g:S\to T$ which is not bijective (it is neither injective nor surjective). The inverse function to $g$ does not exist in this case; graph (e) of a possible candidate, is not the graph of a function.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)