Questions tagged [functional-equation]

Puzzles involving functional equations in mathematics. Use with [mathematics]

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2 answers
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How is this correct? [duplicate]

How is the following equation is correct?$29$ - $1$ = $30$ Hint-
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9 votes
1 answer
214 views

Functional equation: composition to get quadratic

Consider the following functional equation: $$f(f(x))=x^2+x-7\quad\quad\forall\; x\in\mathbb{R}.$$ Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ satisfying this, or not?
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4 votes
3 answers
433 views

Functional inequality?

Find all functions $f:\mathbb{R}\to\mathbb{R}$ s.t. for all $x,y\in\mathbb{R}$, we have $$yf(x)+f(y)\ge f(xy)$$ Problem from the my math olympiad training problem set few weeks before. Functional ...
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6 votes
5 answers
538 views

All wrapped in functions

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)f\big(f(x)+y\big)=f\big(x^2\big)+f(xy)$$ for all $x,y\in\mathbb R$ Problem by me Most elegant solution wins!
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5 votes
2 answers
277 views

What should you substitute?

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$xf(x)-yf(y)=(x-y)f(x+y)$$ for all $x,y\in\mathbb{R}$. Problem by me. Most elegant solution gets the checkmark!
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11 votes
1 answer
855 views

Functional equation by me!

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)f(x+y)=xf(x)+f\big(f(x)\big)f(y)$$ for all $x,y\in\mathbb R$. Source: Problem by me.
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10 votes
4 answers
514 views

Floor functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the following functional equation: $$f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor\quad\quad\text{for all }x,y\in\mathbb{R},$$ where $\lfloor\...
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9 votes
1 answer
373 views

A functional equation again!

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$(x-2)f(y)+f\big(y+2f(x)\big)=f\big(x+yf(x)\big)$$ for all $x,y\in\mathbb{R}$. Source: Math Excalibur Volume 22, Number 1 ...
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10 votes
1 answer
312 views

Functional Equation: Squeeze it

Determine whether there exists a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f\big(x^3+x\big)\le x\le\big(f(x)\big)^3+f(x)$$ for all $x\in\mathbb{R}$. Source: Math Excalibur Volume 22 No....
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9 votes
1 answer
549 views

Functional equation involving 2020th powers

Find all functions $f:\mathbb{N}\to\mathbb{N}$ which satisfy $$ (n-1)^{2020}<\prod_{k=1}^{2020}f^{(k)}(n)<n^{2020}+n^{2019}\quad\quad\text{for all }n\in\mathbb{N}, $$ where $\mathbb{N}$ is the ...
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9 votes
3 answers
437 views

What is $f(2016)$?

Let $f: \mathbb Z \to \mathbb Z $ be a function such that $$f(34f(x)+78)=57$$ What is $f(2016)$? Do we know any other value of $f$ for sure? This is one of my own problems.
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10 votes
3 answers
430 views

Functional equation $f(f(f(x)f(y)))=f(x)f(y^2)$

Find all functions $f: \mathbb R_{>0} \rightarrow \mathbb R_{>0}$ such that $f(f(f(x)f(y)))=f(x)f(y^2)$ for all $x, y \in \mathbb R_{>0}$. I made this problem myself.
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