Mathematics

Problem

You are given that two functions

f

and

g

are bijections satisfying

f^{-1}(x)=\frac{x-1}{3},\qquad g^{-1}(x)=\frac{1}{x+2},

where domains and ranges are chosen so that the expressions are valid. (i) Express

\bigl(f\circ g\bigr)^{-1}(x)

in terms of

f^{-1}

and

g^{-1}

only. (ii) Hence simplify

\bigl(f\circ g\bigr)^{-1}(x)

to a single rational function, stating clearly its domain.