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

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