# Given that f(x)=x-1 and (g\circ f)(x)=3x^2 + 2, find the function of g in similar form? Thank you

Aug 11, 2018

One possibility is $g \left(x\right) = 3 {x}^{2} + 6 x + 5$

#### Explanation:

There are infinite $g$'s which satisfy this condition. Let us consider as $g$ pertaining to the polynomials, for instance let

 g(x) = a x^2+b x+c

then

 g(f(x)) = a (f(x))^2+b f(x) + c = a(x-1)^2+b(x-1)+c

so

 g(f(x)) = (a-3)x^2+(b-2a)x+a+c-b+2 = 3x^2+2

and now comparing coeficients

$a = 3 , b = 6 , c = 5$

hence $g \left(x\right) = 3 {x}^{2} + 6 x + 5$