The functions \(f\) and \(g\) are differentiable. For all \(x\), \(f(g(x))=x\) and \(g(f(x))=x\). If \(f(3)=8\) and \(f'(3)=9\), what are the values of \(g(8)\) and \(g'(8)\) ?

Summary Submit

Since we are given \(f(g(x))=x\) and \(g(f(x))=x\), \(f\) and \(g\) must be inverse functions.

Since \(f(3)=8\), \(\boxed{g(8)=3}\).

We can use the chain rule to obtain the formula for \(g'(x)\):

$$ f(g(x))=x $$ $$ f'(g(x))\cdot g'(x)=1 $$ $$ g'(x)=\frac{1}{f'(g(x))} $$ $$ g'(8)=\frac{1}{f'(g(8))} $$ $$ g'(8)=\frac{1}{f'(3)} $$ $$ \boxed{g'(8)=\frac{1}{9}} $$