Question

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

Answer 1

One possibility is `g(x) = 3x^2+6x+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(x) = 3x^2+6x+5`