Composite Functions

\(\blacksquare\) Function \(f\) maps set \(P\) to set \(Q\), function \(g\) maps set \(Q\) to set \(R\) and function \(gf\) maps set \(P\) to set \(R\).

\(\blacksquare\) Given two functions \(f(x)\) and \(g(x)\), both the functions can be combined and written as \(fg(x)\) or \(gf(x)\) which is defined as \(fg(x)=f[g(x)]\) or \(gf(x)=g[f(x)]\).

\(\blacksquare\) In general, \(fg \ne gf\), \(f^2=ff\), \(f^3=fff\) and so on.