Question

sendo g(x) = x -9 e F[g(x)] = x² + 7 determine F(x)?

Answer 1

f(x) = ax² + bx + c

g(x) = x - 9

f(g(x)) = x² + 7

f(g(x) = f(x-9)

f(x-9) = a . (x-9)² + b . (x - 9) + c
f(x-9) = a . (x² - 18x + 81) + bx - 9b + c
f(x-9) = ax² - 18ax + 81a + bx - 9b + c
f(x-9) = ax² + (b - 18a)x + 81a - 9b + c

x² + 7 = ax² + (b - 18a)x + 81a - 9b + c

a = 1

b - 18a = 0
b - 18.(1) = 0
b - 18 = 0
b = 18

81a - 9b + c = 7
81 - 162 + c = 7
- 81 + c = 7
c = 88

f(x) = x² + 18x + 88