Find 𝐹(𝑥), if 𝑓(𝑥 − 2) = 𝑥3 + 3𝑥2 − 3𝑥 + 7

Answer 1

To find 𝐹(𝑥), we need to substitute (𝑥 + 2) in place of 𝑥 in the equation 𝑓(𝑥 − 2) = 𝑥3 + 3𝑥2 − 3𝑥 + 7:

𝑓(𝑥 − 2) = 𝑥3 + 3𝑥2 − 3𝑥 + 7
𝑓((𝑥 + 2) − 2) = (𝑥 + 2)3 + 3(𝑥 + 2)2 − 3(𝑥 + 2) + 7 (substituting 𝑥 + 2 in place of 𝑥)
𝑓(𝑥) = (𝑥 + 2)3 + 3(𝑥 + 2)2 − 3(𝑥 + 2) + 7

Expanding and simplifying:

𝑓(𝑥) = 𝑥3 + 9𝑥2 + 23𝑥 + 17

Therefore, 𝐹(𝑥) = 𝑥3 + 9𝑥2 + 23𝑥 + 17.