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Question Number 117577 by ZiYangLee last updated on 12/Oct/20

If f(x), g(x) and h(x) are three functions,  where f(x)=2x^5 −8x^2 +1, f(x−3)=g(3x−2)  and g(3x+1)=h(x+3), show h(x)=f(x−5).

$$\mathrm{If}\:{f}\left({x}\right),\:{g}\left({x}\right)\:\mathrm{and}\:{h}\left({x}\right)\:\mathrm{are}\:\mathrm{three}\:\mathrm{functions}, \\ $$$$\mathrm{where}\:{f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{5}} −\mathrm{8}{x}^{\mathrm{2}} +\mathrm{1},\:{f}\left({x}−\mathrm{3}\right)={g}\left(\mathrm{3}{x}−\mathrm{2}\right) \\ $$$$\mathrm{and}\:{g}\left(\mathrm{3}{x}+\mathrm{1}\right)={h}\left({x}+\mathrm{3}\right),\:\mathrm{show}\:{h}\left({x}\right)={f}\left({x}−\mathrm{5}\right). \\ $$

Answered by Olaf last updated on 12/Oct/20

h(x) = g(3(x−3)+1) = g(3(x−2)−2)  = f((x−2)−3) = f(x−5)

$${h}\left({x}\right)\:=\:{g}\left(\mathrm{3}\left({x}−\mathrm{3}\right)+\mathrm{1}\right)\:=\:{g}\left(\mathrm{3}\left({x}−\mathrm{2}\right)−\mathrm{2}\right) \\ $$$$=\:{f}\left(\left({x}−\mathrm{2}\right)−\mathrm{3}\right)\:=\:{f}\left({x}−\mathrm{5}\right) \\ $$

Commented by ZiYangLee last updated on 12/Oct/20

hmmm..

$$\mathrm{hmmm}.. \\ $$

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