Question

If $f:R\to R,f\left(x\right)=2x-3;g:R\to R,g\left(x\right)=x^3+5$, then the value of $\left(fog{)}^{-1}\right(x)$:

• A. ${\left(\frac{x+7}{2}\right)}^{1/3}$
• B. ${(x-\frac{7}{2})}^{1/3}$
• C. ${\left(\frac{x-2}{7}\right)}^{1/3}$
• D. ${\left(\frac{x-7}{2}\right)}^{1/3}$

Answer 1

The correct option is D ${\left(\frac{x-7}{2}\right)}^{1/3}$

$f:R\to R,f\left(x\right)=2x-3$

$g:R\to R,g\left(x\right)=x^3+5$

$\therefore \left(fog\right)\left(x\right)=f\left(g\right(x\left)\right]$

$=f(x^3+5)$

$=2(x^3+5)-3=2x^3+10-3$

$=2x^3+7$

Let $y=\left(fog\right)\left(x\right)=2x^3+7$

$\therefore \left(fog{)}^{-1}\right(y)=x$

$={\left(\frac{y-7}{2}\right)}^{1/3}$

$\therefore \left(fog{)}^{-1}\right(x)={\left(\frac{x-7}{2}\right)}^{1/3}$

So, option (d) is correct.