Question

If the function $f:R\to R$ be defined by $f\left(x\right)=2x-3$ and $g:R\to R$ by $g\left(x\right)=x^3+5$, then find the value of $\left(fog{)}^{-1}\right(x)$.

Answer 1

Given, $f\left(x\right)=2x-3$ and $g\left(x\right)=x^3+5$

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

$=f(x^3+5)$

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

$=2x^3+10-3$

$=2x^3+7$

Now, let $\left(fog\right)\left(x\right)=y$

$⟹2x^3+7=y$

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

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