Expanding a function in terms of basis functions
In the previous section, we introduced some key concepts about decomposing functions, namely that we are breaking down or decomposing a function, 
, into several smaller parts that are easier to interpret and work with. We can also think of those smaller parts as simple building blocks from which we can build up or compose more complicated functions.
Whichever way we choose to look at a function, we need to make the ideas that were introduced in the previous section more concrete. In other words, given a set of simple building block functions, we need to work out how to quantify how much of each of those building block functions are in our function, . To do so, consider the function shown in Eq. 8:
Eq. 8
The coefficient, 
, tells us how much the building block function, 
, contributes to the function, 
. Put another way, 
tells us how much of 
there is in 
. We can use this second way of looking at what 
tells us to work out as...
