Composition of Functions

The things missed in Mathematics!

1.       Here is a process that is very familiar in our day to day life.

2.       It is the process of conversion of same water into ice or vapours.

3.       Here Heating, Refrigeration and Condensation are three different functions. Let us see how these are functions first of all.

4.       Heating changes the water from sets of liquids (water) to sets of gases (vapours) AND Ice to water (sets of solids to sets of liquids) and not vice versa. Hence a function.

5.       Refrigeration changes the water to ice (sets of liquid to sets of gases) and not vice versa. Hence a function.

6.       Condensation changes water vapours to liquid water and not vice versa. Hence a function

7.       Now what will happen if we apply these functions one after another? For example, Lets apply Heating on Ice followed by Heating of water. We get vapours from ice.

8.       Apply Heating of water followed by Condensation. We get water back again.

9.       All these examples are Composition of Functions.

10.   Also note that, the sequence in which these functions are applied does make a difference.

11.   All the sequences may not make sense (e.g. Refrigeration of water and then Condensation).

12.   All the valid sequences may not have an inverse sequence (e.g. Condensation of vapours and then Refrigeration gives us Ice, but Refrigeration of Ice and then Condensation of water is meaningless).

Few formalized representations:

A composition of two functions f and g on a variable x is represented by fog(x) or f(g(x)) and means that first the function g is applied on x and then the function f is applied on the result of g(x).