Here is the question, Suppose f(3k+2)=54k 3 +18k 2 +12k. Nowthe question asks to

Question

Here is the question,
Suppose f(3k+2)=54k3+18k2+12k. Nowthe question asks to find f(x)?
I am really stuck on how to work out the reverse substitution. Aren't there many possible functions that could be the answeranyway? The textbook indicates that there is only ONE answer,though. Please explain fully.
Thanks.
Suppose f(3k+2)=54k3+18k2+12k. Nowthe question asks to find f(x)?
I am really stuck on how to work out the reverse substitution. Aren't there many possible functions that could be the answeranyway? The textbook indicates that there is only ONE answer,though. Please explain fully.
Thanks.

Explanation / Answer

3k + 2 = x -> 3k = x - 2 -> k = (x - 2)/3.
Substituting k = (x - 2)/3, we obtain
f(3k + 2) = f(x) = 54((x - 2)/3)^3 + 18((x - 2)/3)^2 +12((x - 2)/3) =2(x - 2)^3 + 2(x - 2)^2 + 4(x - 2) =(x - 2)(2(x - 2)^2 + 2(x - 2) + 4) = (x - 2)(2(x^2 - 4x + 4) + 2x - 4 + 4) = (x - 2)(2x^2 - 8x + 8 + 2x) =(x - 2)(2x^2 - 6x + 8) =2x^2(x - 2) -6x(x - 2) + 8(x - 2) =2x^3 - 4x^2 - 6x^2 +12x + 8x - 16 =2x^3 - 10x^2 + 20x - 16

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