Decomposing a Composite Function

Find functions \(f\) and \(g\) so that \(f\circ g=F\) when:

1. \(F( x) =\dfrac{1}{x+1}\)
2. \(F( x) =( x^{3}-4x-1) ^{100}\)
3. \(F(t) =\sqrt{2-t}\)

Solution (a) If we let \(f( x) =\dfrac{1}{x}\) and \(g( x) =x+1\), then \[ ( f\circ g) ( x) =f( g( x) ) = \dfrac{1}{g( x) }=\dfrac{1}{x+1}=F( x) \]

(b) If we let \(f( x) =x^{100}\) and \(g( x) =x^{3}-4x-1,\) then \[ ( f\circ g) ( x) =f( g( x) ) =f( x^{3}-4x-1) =( x^{3}-4x-1) ^{100}=F( x) \]

(c) If we let \(f( t) =\sqrt{t}\) and \(g( t) =2-t,\) then \[ ( f\circ g) ( t) =f( g( t) ) =f( 2-t) =\sqrt{2-t}=F( t) \]