# Tagged Questions

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### How to flip a graph over the x-axis but retain the original equation [on hold]

I know this question seems really very basic for this forum, but after about an hour of trying to work it out for myself, I decided to look here for help. The premise I'm trying to create the graph ...
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### Where can I find out more about such functional equations?

Such as this one $$f(x) = x + f(\frac{x}{1-x})$$ or that one $$f(x) = x + f(\frac{1}{1-x})$$ I did some digging on my own: here, here and here but I was curious if there's any existing info on the ...
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### Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...
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### Finding functional equations that a given function satisfies

Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b$ with $a,b \in \mathbb{C}$. I would like to know which functional equations are ...
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### Functions with special separability

Suppose we have differentiable functions $F$, $f_1, \dots, f_n$, and $g_1, \dots, g_n$ satisfy the following relation $$F(x+y) = \sum_{i=1}^n f_i(x) g_i(y).$$ What are the possible forms of $F$?
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### Uniqueness of solutions of functional equations [closed]

A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is $$f(x) = \left( \frac{\beta}{\alpha^{\log_2 \beta}} \right) \alpha^{\log_2 x}.$$ Do we know whether or not the ...
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### Existence of solution for this set of polynomial equations

We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where $$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$ I'm interested in proving that a solution for ...
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### Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$. Our goal is to find $t\in[0,1]$ such that $$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$ Is there a closed-form ...
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### Solve this functional equation with respect to $f$

Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$. My question is: Can we solve this functional equation ...
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