# 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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### Math question functions help me?

I have to find find $f(x,y)$ that satisfies \begin{align} f(x+y,x-y) &= xy + y^2 \\ f(x+y, \frac{y}x ) &= x^2 - y^2 \end{align} So I first though about replacing $x+y=X$ and $x-y=Y$ in the ...
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### The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
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### Help in solving a simple functional equation

I need to find all continuous functions satisfying: $$3f(2x+1)=f(x) + 5x$$ The functional equation looks simple but I am unable to solve it. I tried to convert it into a Cauchy type equation but I ...
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### A functional equation (another)

I would like to find a continuous concave function from $[1/2,1]$ to $[0,1]$ such that $f(1)=1$ and for all $x\in [1/2,1]$ $$f(x)= \frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right).$$ I am ...
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### The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
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### Functional equation: $f(x)f(1/x) = f(x) + f(1/x)$.

If $x \neq 0$ , find $f(x)$ if it satisfies: $f(x)f(1/x) = f(x) + f(1/x)$. I know that the answer is $f(x) = 1 \pm x^n$ where $n \in \mathbb{R}$. I don't know how to show this.
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### Solve the functional equation $\frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right)$

Solve the functional equation $$\frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right),$$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. ...
### Find $f(x)$ where $f(x)(A-\frac{B}{x+B/A})+Cf(x+\frac{B}{A})=0$.
$A, B, C > 0$, $x$ is complex and $Re(x)>0$. My guess is that $f(x)=0$ but I don't know how to prove it.