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