Let $f(x)=x+2$ and $g(x)=x/3.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute \[f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))).\]
+9466 \(f(x)=x+2\) and \(f^{-1}(x)=x-2\)
\(g(x)=x/3\) and \(g^{-1}(x)=3x\)
There may a better way to do this, but here is what I got:
\(\begin{array}\ &&f(g^{-1}(f^{-1}(f^{-1}(g( {\color{Red}f(19)} ))))) \\ =&&f(g^{-1}(f^{-1}(f^{-1}(g( {\color{Red}19+2} )))))\\ =&&f(g^{-1}(f^{-1}(f^{-1}(g( {\color{Red}21} )))))\\ =&&f(g^{-1}(f^{-1}(f^{-1}( {\color{Orange}g( 21 )} ))))\\ =&&f(g^{-1}(f^{-1}(f^{-1}( {\color{Orange}21/3} ))))\\ =&&f(g^{-1}(f^{-1}(f^{-1}( {\color{Orange}7} ))))\\ =&&f(g^{-1}(f^{-1}( {\color{orange}f^{-1}(7)} )))\\ =&&f(g^{-1}(f^{-1}( {\color{orange}7-2} )))\\ =&&f(g^{-1}(f^{-1}( {\color{orange}5} )))\\ =&&f(g^{-1}( {\color{Green}f^{-1}( 5 )} ))\\ =&&f(g^{-1}( {\color{Green}5-2} ))\\ =&&f(g^{-1}( {\color{Green}3} ))\\ =&&f({\color{Blue}g^{-1}( 3 )})\\ =&&f({\color{Blue}3(3)})\\ =&&f({\color{Blue}9})\\ =&&{\color{Fuchsia}f(9)}\\ =&&{\color{Fuchsia}9+2}\\ =&&{\color{Fuchsia}11} \end{array}\)
