The mapping $f:R\to R$ given by $f\left(x\right)=x^3+a{x}^2+bx+c$ is a bijection if

A function $g\left(x\right)$ satisfies the following conditions
(i) Domain of $g$ is $\left(-\infty ,\infty \right)$

(ii)  Range of $g$ is $\left[-1,7\right]$
(iii) $g$ has a period $\pi$ and
(iv) $g\left(2\right)=3$

Then which of the following may be possible.

If $f\left(x\right)={\cot }^{-1}x :R^{+}\to \left(0,\frac{\pi }{2}\right)$

and $g\left(x\right)=2x-x^2 :R\to R.$ Then the range of the function $f\left(g\left(x\right)\right)$  wherever defined is