Question

Let g(x) be the inverse of the function f(x) and $f^{\prime }\left(x\right)=\frac{1}{1+x^3}$ , then f(x) is equal to

• A. $\frac{1}{1+{\left[g\left(x\right)\right]}^3}$
• B. $\frac{1}{1+{\left[f\left(x\right)\right]}^3}$
• C. $1+{\left[g\left(x\right)\right]}^3$
• D. $1+{\left[f\left(x\right)\right]}^3$

Answer 1

The correct option is C $1+{\left[g\left(x\right)\right]}^3$

$f^{\prime }\left(x\right)=\frac{1}{1+x^3}$

Since $g\left(x\right)$ is inverse of function $f\left(x\right)$

then

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

Differentiating $wrt{}^{\prime }{x}^{\prime }$

$f\left(g\right(x\left)\right)=g\left(x\right)=1$

So, $g\left(x\right)=\frac{1}{f\left(g\right(x\left)\right)}$

$\Rightarrow f\left(x\right)=1+\left(g\right(x){)}^3$

Option $\left(C\right)$ is correct