\[\begin{cases} g(x) = x^2 - 11x + 30 \\ g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42 \end{cases}\]
Let \(g\) and \(f\) be the monic polynomial functions that satisfy the system above. What is the value of \(f(7)\)?
+128406 f(x) must be a quadratic just like g(x)
So
(ax^2 + bx + c)^2 - 11(ax^2 + bx + c) + 30 =
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
a must =1
And
2ab must = 2(1)b = -14 ⇒ b = -7
And
(2ac + b^2 - 11a) = 62
2c + 49 - 11 = 62
2c + 38 = 62
2c = 24
c =12
Check
2bc -11b = -91 ??? c^2 - 11c + 30 = 42 ???
2(-7)(12) -11(-7) 12^2 - 11(12) + 30 =
-168 + 77 144 - 132 + 30 =
-91 42
So
f(x) = x^2 - 7x + 12
And
f(7) = 7^2 - 7(7) + 12 = 12
