Question

If an invertible function $f\left(x\right)$ is defined as $f\left(x\right)=3x-2,g\left(x\right)$ is also an invertible function such that $f^{-1}\left(g^{-1}\left(x\right)\right)=x-2,$ then $g\left(x\right)$ is

• A. $\frac{x-8}{3}$
• B. $\frac{x+8}{3}$
• C. $\frac{x-3}{8}$
• D. $\frac{x+3}{8}$$3\left(x-2\right)-2=g^{-1}\left(x\right)$

Answer 1

The correct option is B

$\frac{x+8}{3}$

Finding the value:

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

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

$\Rightarrow$ $3\left(x-2\right)-2=g^{-1}\left(x\right)$

$\Rightarrow$ $g^{-1}\left(x\right)=3x-8$

Put, $x=g\left(x\right)$,

$\Rightarrow$ $g^{-1}\left(g\left(x\right)\right)=3g\left(x\right)-8$

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

Hence, correct answer is option $\left(B\right)$