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zmudz
 one year ago
Let F(x) be the realvalued function defined for all real x except for x = 0 and x = 1 and satisfying the functional equation \(F(x) + F((x1)/x) = 1+x.\)Find the F(x) satisfying these conditions.
Write F(x) as a rational function with expanded polynomials in the numerator and denominator.
Help needed! Thank you!
zmudz
 one year ago
Let F(x) be the realvalued function defined for all real x except for x = 0 and x = 1 and satisfying the functional equation \(F(x) + F((x1)/x) = 1+x.\)Find the F(x) satisfying these conditions. Write F(x) as a rational function with expanded polynomials in the numerator and denominator. Help needed! Thank you!

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\[F(x)+F\left(\frac{x1}{x}\right) =1+x\tag{1}\] replace \(x\) by \((x1)/x\) in \((1)\) and get \[F\left(\frac{x1}{x}\right)+F\left(\frac{1}{x1}\right) =1+\frac{x1}{x}\tag{2}\] replace \(x\) by \(1/(x1)\) in \((1)\) and get \[F\left(\frac{1}{x1}\right)+F\left(x\right) =1+\frac{1}{x1}\tag{3}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2If you let \(F(x)=a\), \(F\left(\frac{x1}{x}\right)=b\) and \(F\left(\frac{1}{x1}\right)=c\), the previous equations become : \[a+b = 1+x\tag{1'}\] \[b+c = 1+\frac{x1}{x}\tag{2'}\] \[c+a = 1+\frac{1}{x1}\tag{3'}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.23 equations and 3 unknowns (a,b,c) you can solve them
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