# Tagged Questions

The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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### Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$

Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$. We have $f(0) = 0$ and $f(x) = 2f(2x) - x$, but I am not sure how to convert this functional equation into something ...
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### Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
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### There is no function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$

Problem: Prove that there is no differentiable function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$. I could not make much progress, except for observing that any derivatives (any ...
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### Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
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### Is the gamma function a solution to this problem?

A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense ...
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### Functional Equation $f(f(x))=af(x)+bx$

For real numbers $a$ and $b \neq 0$, $f(x)$ satisfies the following: $$f(f(x))=af(x)+bx$$ (1) $f(x)$ is continuous and $0<a, b<\frac{1}{2}$, show that the equation $f(x)=x$ has a real root, ...
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### Functional equation $g(2x )= 1/2 g(x)$

I am trying to solve functional equation for $g: \ (0, \infty) \mapsto ( 0, \infty)$ $$g( 2x ) =\frac 12 g(x)$$ Wolfram claims, and it is intuitive, that the function is $g(x) = C \frac{1}{x}$. But ...
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### If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Assume $f: (0, \infty) \to \mathbb{R}$ is a continuous function such that for any $x,y > 0$, $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$. Find $f$. I would work with each condition ...
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### Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$. We have that $f(0) = 0$ and $f(x+1) = f(x)+f(1)+2x$ and thus $f(x+1) - f(x) = f(1)+2x$. Then we see ...
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### Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations

Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations: $f(x)g(y) = x+y$ and $f(x) + g(y) = xy$. I think we should ...
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### If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$ I find it hard ...
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### Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$. We know ...
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### Can I prove that $f(x,y)$ can be written as $g(x+y)$ under certain conditions.

I have $f(x,y):R^2\rightarrow R$. I know $f(x,y)=f(y,x)$ and $f(x+d,y)=f(x,y+d)$. Can I prove that I can express $f(x,y)$ as $g(x+y)$. This is where I got: $f(x+d,y)=f(x,y+d)$, I plug in $x=0$ Gives ...
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### Finding a function which satisfies an equation

During my analysis course, I found this interesting problem: Let $f: A \to A$ a function such that $f(f(x))+3x=4f(x)$ for every $x \in A$, where A is a finite set of positive real numbers. Find the ...
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### Functional equation $(n-1)^2 < f(n) f(f(n)) < n^2 +n$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$(n-1)^2 < f(n) f(f(n)) < n^2 +n$$ for all $n \in \mathbb{N}$.
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### Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$:$f(f(f(x)+y)+y)=x+y+f(y)$

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$: $f(f(f(x)+y)+y)=x+y+f(y)$ I got the following: (1)$f$ is injective (2) $f(0)=0$ (3)$f(f(f(x)))=x$ But then ...
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### Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant

Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant I was trying to divide into 3 cases: when f(x) has a root, when f(x)>0 and when f(x)<0 .. But I ...
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### Help finding a $C^{1}$ function, with given $C^{1}$ functions, a relation, and some additional assumptions.

I'm down to the following problem (see below) that I just need some insight on (I couldn't find anything close to help online, via other posts here, etc.). My initial thoughts were to use the Inverse ...
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### Searching a formula for scaling/mapping a variable based on 3 known values

I am sending an specc'ed integer (X) between -2048, and 2048 to a synthesizer to control its tuning. When X is 0 = Tuning on Synth is 440 (default) When X is 2048 = Tuning on Synth is 546.42 When X ...
$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.