If g(x)=x2+x-2 and (1/2)(g o f)(x)=2 x2-5 x+2 then one such function f(x)=

Question

If g(x)=x2+x-2 and (1/2)(g o f)(x)=2 x2-5 x+2 then one such function f(x)= (A) 2 x-3 (B) 2 x+3 (C) 2+2 x (D) 2 x2-3 x-1

Tardigrade Admin · Accepted Answer

Given, g(x)=x2+x-2 and (1/2)(g o f)(x)=2 x2-5 x+2 ⇒ g(f(x))=4 x2-10 x+4 ⇒ (f(x))2+(f(x))-2=4 x2-10 x+4 Now, it is necessary that f(x) should be linear polynomial expression, so let f(x)=a x +b, then (a x +b)2+(a x +b)-2=4 x2-10 x+4 ⇒ a2 x2+(2 a b +a) x+(b2+b-2) =4 x2-10 x+4 On comparing the coefficient of different kind of terms, we are getting a2=4 ⇒ 2 a b +a=-10 and b2+b-2=4 So, a=± 2, then b = begincases -3, textif a=2 2, textif a-2 endcases and these value satisfy the all above relations, so f(x)=2 x-3 text or -2 x+2

Q. If $g (x) = x^{2} + x - 2$ and $\frac{1}{2} (g o f) (x) = 2 x^{2} - 5 x + 2$ then one such function $f (x) =$

$2 x - 3$

$2 x + 3$

$2 + 2 x$

$2 x^{2} - 3 x - 1$

Q. If g(x)=x2+x−2 and 21​(gof)(x)=2x2−5x+2 then one such function f(x)=

2x−3

2x+3

2+2x

2x2−3x−1

Q. If $g (x) = x^{2} + x - 2$ and $\frac{1}{2} (g o f) (x) = 2 x^{2} - 5 x + 2$ then one such function $f (x) =$

$2 x - 3$

$2 x + 3$

$2 + 2 x$

$2 x^{2} - 3 x - 1$