g(x) = x^2 - 11x + 30
g(f(x)) = x^4 - 8x^3 + 11x^2 + 20x + 6
Let g and f be the monic polynomial functions that satisfy the system above. What is the value of f(7)?
+129845 Let f(x) = ax^2 + bx +c
g(f(x)) = (ax^2 + bx + c)^2 - 11(ax^2 + bx + c) + 30
g (f(x)) = a^2 x^4 + 2 a b x^3 + 2 a c x^2 + b^2 x^2 + 2 b c x + c^2 - 11ax^2 - 11bx - 11c + 30
g(f(x)) = a^2x^4 + 2abx^3 + (2ac + b^2 - 11a)x^2 + (2bc - 11b)x + (c^2 - 11c + 30)
And g (f(x)) = x^4 -8x^3 + 11x^2 + 20x + 6
Equating coefficients
a = 1 or -1
b = - 4 or 4
Let a =1 and b = -4
(2ac + b^2 - 11a) = 11 → 2c + 16 - 11 = 11 → c = 3
So
f(x) = x^2 - 4x + 3
And
f(7) = 7^2 - 4(7) + 3 = 24
