$\text{ If }f\left(x\right)=2x^3+7x-5\text{ , then }{f}^{-1}\left(4\right)\text{ is }$

$\text{ Note that }f\text{ is bijective, hence }{f}^{-1}\text{ exist when }y=4\text{ , }$

$\begin{array}{l}2x^3+7x-9=0\Rightarrow 2x^2(x-1)+2x(x-1)+9(x-1)=\\ \Rightarrow (x-1)\left(2x^2+2x+9\right)=0\end{array}$

$\Rightarrow x=1\text{ only as }2x^2+2x+9=0\text{ has no other real roots ,since ∆<0}$ 

$\Rightarrow f\left(1\right)=4 \Rightarrow f^{-1}\left(4\right)=1$