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Question Number 12591 by @ANTARES_VY last updated on 26/Apr/17
This  ∮(x)=3−(x^2 /(x^4 +3x^2 +1))  find  the  sphere  of  function  values
$$\boldsymbol{\mathrm{This}} \\ $$$$\oint\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{3}−\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}}\:\:\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{sphere}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{function}}\:\:\boldsymbol{\mathrm{values}} \\ $$
Commented by ajfour last updated on 26/Apr/17
∮(x)=3−(1/((x+(1/x))^2 +1))    (if x≠0)  at x=1 , ∮(x) has least value       =3−(1/5)  whenx→±∞, ∮(x) →3  and if  x=0 , ∮(x)=3  so range of ∮(x) is [((14)/5),3] .
$$\oint\left({x}\right)=\mathrm{3}−\frac{\mathrm{1}}{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} +\mathrm{1}}\:\:\:\:\left({if}\:{x}\neq\mathrm{0}\right) \\ $$$${at}\:{x}=\mathrm{1}\:,\:\oint\left({x}\right)\:{has}\:{least}\:{value} \\ $$$$\:\:\:\:\:=\mathrm{3}−\frac{\mathrm{1}}{\mathrm{5}} \\ $$$${whenx}\rightarrow\pm\infty,\:\oint\left({x}\right)\:\rightarrow\mathrm{3} \\ $$$${and}\:{if}\:\:{x}=\mathrm{0}\:,\:\oint\left({x}\right)=\mathrm{3} \\ $$$${so}\:{range}\:{of}\:\oint\left({x}\right)\:{is}\:\left[\frac{\mathrm{14}}{\mathrm{5}},\mathrm{3}\right]\:. \\ $$

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