Question

If $f:R\to R$ and $g:R\to R$ are defined by $f\left(x\right)=3x-4$, and $g\left(x\right)=2+3x$, find $\left(g^{-1}o{f}^{-1}\right)\left(5\right)$.

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{5}$

Answer 1

The correct option is C $\frac{1}{3}$

Given, $f\left(x\right)=3x-4$

Let $y=3x-4$

$\Rightarrow x=\frac{y+4}{3}\Rightarrow f^{-1}\left(x\right)=\frac{x+4}{3}$

Also given $g\left(x\right)=2+3x$

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

Thus $g^{-1}\left({\text{of}}^{-1}\right)\left(x\right)=\frac{\left(\frac{x+4}{3}\right)-2}{3}$

$\Rightarrow g^{-1}\left({\text{of}}^{-1}\right)\left(x\right)=\frac{x-2}{9}\Rightarrow g^{-1}\left({\text{of}}^{-1}\right)\left(5\right)=\frac{5-2}{9}=\frac{1}{3}$

So, option C is correct.