$\text{ If }g\left(x\right)=x^2+x-2\text{ and }\frac{1}{2}\left(\text{ gof }\right)\left(x\right)=2x^2-5x+2\text{ , then }f\left(x\right)\text{ is equal to }$

$g\left(x\right)=x^2+x-2 and \left(gof\right)\left(x\right) =4x^2-10x+4$

             $\Rightarrow g\left(f\left(x\right)\right)={\left(f\left(x\right)\right)}^2+f\left(x\right)-2 = 4x^2-10x+4$

            ${\left(f\left(x\right)\right)}^2 +f\left(x\right)-\left(4x^2-10x+6\right)=0\Rightarrow$

          $f\left(x\right)=\frac{-1\pm \sqrt{16x^2-40x+25}}{2}=\frac{-1\pm \left(4x-5\right)}{2} =2x-3 or -2x+2$  

         $hence f\left(x\right)=2x-3$