# 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
2 years, 3 months ago

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You can put $$x=1$$ so that mean $$f(f(y))=f(y)+1$$ which mean $$f(x)=x+1$$

- 2 years, 2 months ago

I got that .

I want to know if there is another function satisfies this

- 2 years, 2 months ago

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.

- 2 years, 2 months ago