Question Number 128132 by I want to learn more last updated on 04/Jan/21
$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{x}}\right)\:\:\:=\:\:\:\mathrm{x}^{\mathrm{4}} \:\:+\:\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }\:,\:\:\:\:\:\mathrm{find}\:\:\mathrm{f}\left(\mathrm{5}\right) \\ $$
Answered by Olaf last updated on 04/Jan/21
![f(x+(1/x)) = x^4 +(1/x^4 ) = (x^2 +(1/x^2 ))^2 −2 = [(x+(1/x))^2 −2]^2 −2 If u = x+(1/x) f(u) = (u^2 −2)^2 −2 f(5) = (25−2)^2 −2 = 527](Q128133.png)
$${f}\left({x}+\frac{\mathrm{1}}{{x}}\right)\:=\:{x}^{\mathrm{4}} +\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:=\:\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} −\mathrm{2} \\ $$$$=\:\left[\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} −\mathrm{2}\right]^{\mathrm{2}} −\mathrm{2} \\ $$$$\mathrm{If}\:{u}\:=\:{x}+\frac{\mathrm{1}}{{x}} \\ $$$${f}\left({u}\right)\:=\:\left({u}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2} \\ $$$${f}\left(\mathrm{5}\right)\:=\:\left(\mathrm{25}−\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\:=\:\mathrm{527} \\ $$