Question

Given the function f(x) = x + 3 and g(x) = a + bx2. If gf(x) = 6x2 + 36x + 56,

find the value of a and of b.​

Answer 1

`g(f(x))=a+b\cdot(x+3)^2=a+b(x^2+6x+9)=a+bx^2+6bx+9b=\\=bx^2+6bx+9b+a\\\\6x^2+36x+56=bx^2+6bx+9b+a\\b=6\\9b+a=56\\9\cdot6+a=56\\a=2\\\\\\\boxed{a=2,b=6}`

Answer 2

a = 2, b = 6

Explanation:

To obtain g(f(x)) substitute x = f(x) into g(x), that is

g(x + 3) = a + b(x + 3)²

= a + b(x² + 6x + 9) = a + bx² + 6bx + 9b

For a + bx² + 6bx + 9b = 6x² + 36x + 56

Then coefficients of like terms must be equal

Comparing like terms

x² term ⇒ b = 6

constant term ⇒ a + 9b = 56 ⇒ a + 54 = 56 ⇒ a = 56 - 54 = 2