Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 117934 by Ar Brandon last updated on 14/Oct/20

Let f : [1,∞)→[2,∞) be the function defined by f(x)=x+(1/x) If g : [2,∞)→[1,∞), is a function such that (g○f)(x)=x for all x≥1. Show that g(t)=((t+(√(t^2 −4)))/2)

$$\mathrm{Let}\:{f}\::\:\left[\mathrm{1},\infty\right)\rightarrow\left[\mathrm{2},\infty\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)={x}+\frac{\mathrm{1}}{{x}} \\ $$$$\mathrm{If}\:\mathrm{g}\::\:\left[\mathrm{2},\infty\right)\rightarrow\left[\mathrm{1},\infty\right),\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:\left(\mathrm{g}\circ{f}\right)\left({x}\right)={x} \\ $$$$\mathrm{for}\:\mathrm{all}\:{x}\geqslant\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{g}\left({t}\right)=\frac{{t}+\sqrt{{t}^{\mathrm{2}} −\mathrm{4}}}{\mathrm{2}} \\ $$

Answered by Lordose last updated on 14/Oct/20

g(f(x))=x g(x+(1/x)) = x x+(1/x)= y x^2 −yx + 1=0 x^2 −yx +(−(y/2))^2 = −1 + (y^2 /4) (x−(y/2))=±(√((y^2 −4)/4)) x= (1/2)(y±(√(y^2 −4))) g(y)= (1/2)(y±(√(y^2 −4))) g(t) = (1/2)(t±(√(t^2 −4)))

$$\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{x} \\ $$$$\mathrm{g}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:\mathrm{x} \\ $$$$\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}=\:\mathrm{y} \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{yx}\:+\:\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{yx}\:+\left(−\frac{\mathrm{y}}{\mathrm{2}}\right)^{\mathrm{2}} =\:−\mathrm{1}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\left(\mathrm{x}−\frac{\mathrm{y}}{\mathrm{2}}\right)=\pm\sqrt{\frac{\mathrm{y}^{\mathrm{2}} −\mathrm{4}}{\mathrm{4}}} \\ $$$$\mathrm{x}=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{y}\pm\sqrt{\mathrm{y}^{\mathrm{2}} −\mathrm{4}}\right) \\ $$$$\mathrm{g}\left(\mathrm{y}\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{y}\pm\sqrt{\mathrm{y}^{\mathrm{2}} −\mathrm{4}}\right) \\ $$$$\mathrm{g}\left(\mathrm{t}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{t}\pm\sqrt{\mathrm{t}^{\mathrm{2}} −\mathrm{4}}\right) \\ $$

Commented by Ar Brandon last updated on 14/Oct/20

Thank You ��

Terms of Service

Privacy Policy

Contact: info@tinkutara.com