# 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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### Fuzzy's Involution Generalization

As a consequence from my personal attempt to generalize the Fuzzy Logic's negation, I've got the following functional equation: \begin{align*} &f(x) + f(\alpha + \beta - x) = 2*f\left(\frac{\alpha+...
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### Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
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### Proving a function to be a difference

I am concerned with a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(f(x_1,x_2),f(x_3,x_2))=f(x_1,x_3)$$ for any $x_1,x_2,x_3\in \mathbb{R}$, and $$f(x,0)=x$$ $$f(x,x)=0$$ ...
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### Functional differential equation separatrix

I've been spinning my wheels with the following differential equation, and would greatly appreciate any guidance on ways to attack it. I have $u(x) \geq 0$ for all $x$. Further, $x \geq 0$. The ...
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### Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$\forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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### Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f\left(x(y+1)\right)+y=xf(y)+f\left(x+g(y)\right)$

Find all pairs of functions $f,g: \mathbb R \rightarrow \mathbb R$ such that 1). $f\left(xy(y+1)\right)+y=xf(y)+f\left(x+g(y)\right)$ for all real $x,y$; 2). $f(0)+g(0)=0.$ My work so far:...
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### Square root of a function (in the sense of composition)

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can ...
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### Let $f$ be a continuous and positive function on $\mathbb{R}_{+}$ such that $\lim_{x \to \infty} <1$

Let $f$ be a continuous and positive function on $\mathbb{R}_{+}$ such that $\displaystyle\underset{x \to \infty}{\lim} \frac{f(x)}{x} <1$. Prove the equation $$f(x)=x$$ has at least one solution ...
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### Problem based on $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$

Let $$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$$ for all real $x$ and $y$. Also, $f'(0) = -1$ and $f(0)=1$. What is then the value of $f(2)$? I got it as $-1$ by using some algebraic ...
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### Converting parametric function into cartesian

I am trying to convert the parametric function $x(t) = a\cdot(t - \sin(t)) + b\cdot\cos\left(\frac{t}{2}\right)$ $y(t) = a\cdot\cos(t) + b\cdot\sin\left(\frac{t}{2}\right)$ into a cartesian form. ...
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### How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$. [duplicate]

I have a question concerning the proof of the continuity of $f(x) = \ln x$. I read in a comment by Pedro Tamaroff to ncmathsadist's answer to this question that this can be proved in two steps: ...
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### How do I determine periodicity of a function through a system of functional equations?

I was given these equations, $f(k+x) = f(k-x)$, $f(2k+x)= -f(2k-x)$ . k is assumed a constant. I was asked to comment whether $f(x)$ is even or odd. By solving I came to the equation, $f(-x)=-f(x)$,...
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### Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.