Composite functions

Summary

• Given two functions \(g: A→B\) and \(f: B→C\) where the domain of \(f\) is the same as the codomain of \(g\) we can define a new function \(h: A→C\) called the composite of \(f\) and \(g\) by

\[y=h(x)=f\left( g\left( x \right) \right)\]

• \(h\) is denoted by \(f∘g\) , that is, \((f∘g)(x)=f\left( g\left( x \right) \right)\) .
• \(f\) is called the outer or exterior function and \(g\) the inner or interior function .
• If we denote \(u=g(x)\) such that \(h(x)=f(u)\)
• Example: \(y=h(x)=exp \left( x^2 \right)\) then \(h=f∘g\) where \(u=g(x)=x^2\) and \(y=f(u)=e^u\) .