Question

If $f$ and $g$ are invertible functions given by $f\left(x\right)=3x-2$ and $\left(gof{)}^{-1}\right(x)=x-2$, then the value of $g\left(7\right)$ is

Answer 1

$f\left(x\right)=3x-2\Rightarrow f^{-1}\left(x\right)=\frac{x+2}{3}$
We know that,
$\left(gof{)}^{-1}\right(x)=f^{-1}o{g}^{-1}(x)$
Now,
$\left(gof{)}^{-1}\right(x)=x-2\Rightarrow f^{-1}o{g}^{-1}(x)=x-2\Rightarrow f^{-1}(g^{-1}\left(x\right))=x-2\Rightarrow \frac{g^{-1}\left(x\right)+2}{3}=x-2\Rightarrow g^{-1}(x)=3x-8\Rightarrow g(x)=\frac{x+8}{3}\therefore g(7)=5$