Question

Let $f:R\to R,g:R\to R$ be two function such that
$f\left(x\right)=2x-3,g\left(x\right)=x^3+5$
The function $\left(fog{)}^1\right(x)$ is equal to.

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

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

$\left(fog\right)\left(x\right)=fg\left(x\right)=f(x^3+5)=2(x^3+5)-3=2x^3+7$

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

$y-7=2x^3$

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

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