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 Question from Joshua: Let g(x)=(x)^(2)+x-1 Find such a function f such that (fog)(x)=(x)^(4)+2(x)^(3)-3(x)^(2)-4x+6

Hi Joshua,

You want $f(g(x)) = x^4 + 2x^3 - 3x^2 -4x + 6.$ This is a fourth degree polynomial and $g(x)$ is a second degree polynomial so I wondered how close is $g(x)^2$ to $f(g(x))?$ I calculated $g(x)^2$ and then looked at $f(g(x)) - g(x)^{2}.$ This is a second degree polynomial with leading term $4 x^{2}.$ I then wondered how close is $f(g(x)) - g(x)^{2}$ to $4 g(x)?$

I hope this helps,
Penny

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