AP EAMCET Engineering 2018 Apr 23 Shift 2 Paper

Consider the expression,
g(x)=x2+x−2
And,

1

2

(gοf)(x)=2x2−5x+2
Then,

1

2

(gοf)(x)=4x2−10x+4
And,
(f(x)2)+f(x)−2=4x2−10x+4
For the function to be linear f(x)=ax+b so,
(ax+b)2+(ax+b)−2 =4x2−10x+4
a2x2+(2ab+a)x+(b2+b−2) =4x2−10x+4
By the comparison of coefficients,
a2=4
And,
2ab+a=−10
And,
b2+b−2=4
So,
a±2
Then,
b={

−3;	if‌a=2
2;	if‌a=−2

These values satisfies above relations, thus,
f(x)=2x−3‌or−2x+2