# Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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### Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
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### A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
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### Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$

If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that : $$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$ My attempt ...
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### Additive and Bijective function on the real line

I was trying to solve the following functional equation: $f(f(x-y))=f(x)-f(y)$. And I concluded that $f$ must be additive and bijective. The question is: Let $f:\mathbb{R} \to \mathbb{R}$ be an ...
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### Finding f$(x)$ from a given functional equation [on hold]

Find all continuous functions $f:(0,1)\rightarrow\mathbb{R}$ such that $$f\left(\dfrac{x-y}{\ln x-\ln y}\right)=\dfrac{f(x)+f(y)}{2},$$ and $x\neq y$. I tried by solving putting $y=\frac{x}{2}$ but ...
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### Cauchy's functional equation from complex to reals

I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (...
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### Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R}$ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R}$ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y$ by setting $x=0$ . By this equation we ...