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Functional Equation

I don't know how to solve functional equations .

How to find functions \( f : \mathbb R^+ \to \mathbb R^+ \) that satisfy the functional equation

\[ f(x.f(y)) = f(xy) + x \]

Note by Cr Ãnänd
10 months, 2 weeks ago

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You can put \(x=1\) so that mean \(f(f(y))=f(y)+1\) which mean \(f(x)=x+1\) Kaito Einstein · 10 months, 2 weeks ago

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@Kaito Einstein I got that .

I want to know if there is another function satisfies this Cr Ãnänd · 10 months, 2 weeks ago

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@Cr Ãnänd No , i don't think so,

we have that \(f(f(x)f(y))=(f(f(x)y)+f(x))=f(f(x)f(y))+y+f(x)\)

and also \(f(f(x)f(y))=f(f(x)f(y))+x+f(y)\)

which mean \(f(f(x)f(y))+y+f(x)=f(f(x)f(y))+x+f(y)\)

so \(f(x)-x=f(y)-y=k\)

so \(f(x)=x+k\)

we substitute the result in the first equation we find \(k=1\).

Hope that help. Kaito Einstein · 10 months, 2 weeks ago

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