Question

If $f,g$ are invertible functionsgiven by $f\left(x\right)=3x-2$ and $\left(gof{)}^{-1}\right(x)=x-2$, then

• A. $g\left(x\right)=\frac{x+8}{2}$
• B. $g\left(x\right)=\frac{x-8}{3}$
• C. $g\left(x\right)=\frac{x+8}{3}$
• D. $g\left(x\right)=\frac{2x+8}{3}$

Answer 1

The correct option is C $g\left(x\right)=\frac{x+8}{3}$

$f\left(x\right)=3x-2$
$\Rightarrow f^{-1}\left(x\right)=\frac{x+2}{3}$
We know that,
$\left(gof{)}^{-1}\right(x)=f^{-1}o{g}^{-1}(x)$
So, $\left(gof{)}^{-1}\right(x)=x-2$
$\Rightarrow f^{-1}o{g}^{-1}\left(x\right)=x-2$
$\Rightarrow f^{-1}\left(g^{-1}\right(x\left)\right)=x-2$
$\Rightarrow \frac{g^{-1}\left(x\right)+2}{3}=x-2$
$\Rightarrow g^{-1}\left(x\right)=3x-8$
Inverse function for $g^{-1}\left(x\right)=g\left(x\right)=\frac{x+8}{3}$