# Questions tagged [functional-equation]

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

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### A functional equation: Composition to get a... linear function? [closed]

Suppose we have a function such that$$f(f(x))=2x+4\quad\forall x\in\mathbb R$$Does such a function $f:\mathbb R\to\mathbb R$ exist that satisfies the relation, or does such a function not exist?
• 2,241
5 votes
1 answer
309 views

### Function that is 0 for all positive integers divisible by x and 1 otherwise

I am working on dice probabilities and I need a function where every xth item of the set of positive integers > 0 (n) is 0 and 1 otherwise.* x is also a positive integers. So, can you, without the ...
• 161
0 votes
2 answers
148 views

### How is this correct? [duplicate]

How is the following equation is correct?$29$ - $1$ = $30$ Hint-
10 votes
1 answer
232 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?
• 117k
4 votes
3 answers
454 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 ...
• 5,600
6 votes
5 answers
547 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!
• 5,600
5 votes
2 answers
280 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!
• 5,600
11 votes
1 answer
866 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.
• 5,600
10 votes
4 answers
528 views

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