Question

If $f\left(x\right)=3x-2$ and $\left(gof{)}^{-1}\right(x)=x-2$, then find the function $g\left(x\right)$

Answer 1

$f\left(x\right)=3x-2$

${(g.f)}^{-1}=x-2$

Let $y=x-2\Rightarrow x=2+y\Rightarrow y^{-1}=2+x$

${\left({(g.f)}^{-1}\right)}^{-1}=2+x$

$\Rightarrow g.f=2+x$

$\Rightarrow g\left(f\left(x\right)\right)=x+2$

$\Rightarrow g(3x-2)=3x-2+2=3x$

Let $3x-2=t\Rightarrow x=\frac{t+2}{3}$

$g\left(t\right)=\frac{t+2}{3}+2$

$g\left(t\right)=\frac{t+8}{3}$

$\therefore g\left(x\right)=\frac{x+8}{3}$