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# Solving a system of functional equations

## Question

h(f(x)) + g(h(x)) + f(g(x)) = 2x^2 + 11x + 14

f(h(x)) + h(g(x)) + g(f(x)) = 2x^2 - 15x + 66

f(g(x)) = g(f(x))

h(g(x)) - g(h(x)) = -16x + 72

h(f(x)) + f(h(x)) = 2x^2 + 10x  + 30

f(x) * g(x) = h(x) - 3x - 40

-----
f(x) =
g(x) =
h(x) =

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Spoiler

hf+gh+fg-fh-hg-gf = 26x - 52

hf+gh-fh-hg = 26x - 52

hf-fh = 26x - 52 - 16x + 72 = 10x + 20

2hf = 10x + 20 + 2x^2 + 10x + 30 = 2x^2 + 20x +50
-2fh = 10x + 20 - 2x^2 - 10x - 30 = -2x^2 - 10

hf = x^2 + 10x + 25 = (x+5)^2
fh = x^2 + 5

seems like...
f(x) = x+5
h(x) = x^2

(x + 5) * g(x) = x^2 - 3x - 40
(x + 5) * g(x) = (x-8)*(x+5)
g(x) = x-8

check...
fg = x-3 = gf

(x^2 + 10x + 25) + g(h(x)) + (x-3) = 2x^2 + 11x + 14
gh = 2x^2 + 11x + 14 - (x^2 + 11x + 22) = x^2 - 8

(x^2 + 5) + h(g(x)) + (x-3) = 2x^2 - 15x + 66
hg = 2x^2 - 15x + 66 - (x^2 + x + 2) = x^2 - 16x + 64 = (x-8)^2

h(g(x)) - g(h(x)) = -16x + 72

Looks like it checks out.

f(x) = x+5
g(x) = x-8
h(x) = x^2

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