Thursday 10 September 2009

Maths - Functions (Part 2)

Composition of Functions


Say we have 2 functions:

  • fR ------> S1

  • gS2 -----> T




These functions are only composable, or have an inter-relationship when

  • S1S2


When these conditions are met, we can safely define the composite of the functions as:

  • hg о f


This states that h is equal to f followed by g. This definition can also be written as:

  • h(x) = g(f(x))


Therefore, we can see that g о f has source, or domain, R (which is the source of f), and has target, or co-domain, T. (which is the target of g). Therefore, we can now write:

  • g о f :  R -------> T


As you can see, the intermediate domain S is not detectable from the composite of this function.

OK, now consider the following 3 functions:

  • fS -------> L

  • gL ------> R

  • hR ------> T


These can be composed is various ways. So we can form:

  • g о f: S -------> R

  • h о g: L -------> T


These may then be further composed, to achieve:

  • h о (g о f)S -------> T

  • (h о g) о fS -------> T


This shows us that composition of functions is associative, meaning we can remove the brackets and write:

  • h о g о f


Just to note, it is also possible to sometimes omit the ‘о’ and just right:

  • hgf


The same can be done for the above functions as well. And thats it, end of Functions :)

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