begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §3. FUNCTIONS. "end quote
These pages assume that the reader has the book and is reading along.
topic index     contents
Please read these notes from the beginning to here and then paragraph 6 of §3.

COMPOSITION OF FUNCTIONS

begin quote ita-1-3-6-pg9 "
If   f : XY   and   g : YZ   are functions, one can define the composition of f and g, or composed function, a function from   X   into   Z   by associating to each element of   X   and element of   Z   in the obvious way: given an element of   X,   one first uses   f   to get an element of   Y,   then one uses   g   to get from this last element an element of   Z.   The composed function is usually denoted   g f,   so that we have   g f : XZ,   with   ( g f )(x) = g( f (x))   for each   .
"end quote

This composition of functions is a very simple idea, but I find that it is also confusing. We mention the function   f   before the function   g   and we call it the composition of   f   and   g,   but the symbol   g f   has   g   first. This last fact, however, is a memory device for the composition's value:   ( g f )(x) = g( f (x)).   The parentheses in this expression will also be a source of confusion for those who are not familiar with standard mathematical notation. The first set of parentheses indicate that   g f   is the symbol or name of the composed function, just as   f   and   g   are the names or symbols of functions. All three other pairs of parentheses enclose the arguments of the functions. Finaly, I would like to look at problem 6 on page 12:

begin quote ita-1-Problem-6-pg12 "
Prove that if   f : XY,   g : YZ   and   h : ZW   are functions, then   h ( g f )   =   ( hg ) f.
"end quote

Let the proof be an exercise, and please see the example here.