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Composing Functions

A function tells you what to do with an argument.
If  
\[f(x)=3x+2\]
,  
\[x\]
  is the argument and  
\[f\]
  is the function which says ' multiply the argument by three then add two'.
Looking at functions this way is especially useful when composing functions.
Suppose  
\[f(x)=3x+2, \: g(x)=x^2+x\]

\[f(g(x))\]
  means 'take the argument  
\[g(x)\]
, multiply it by three then add two'.
\[f(g(x))=3g(x)+2=3(x^2+x)+2=3x^2+3x+2\]

and  
\[f(g(1))=3g(1)+2=3(1^2+1)+2=8\]

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