Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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3
votes
0answers
30 views

Functional equation for cosine on $[0,\frac{\pi}{2}]$

This was part of last week's homework assignment in my Clac 1 class: Let $f:[0,\frac{\pi}{2}]\rightarrow \mathbb{R}^+\cup\{0\}$ be continuous, $$f(\frac{\pi}{2})=0 \hspace{2mm} \textrm{and} \hspace{...
3
votes
0answers
53 views

$(x-y)(f(f(x)^2)-f(f(y)^2))=(f(x)-f(y))(f(x)^2-f(y)^2$

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies: $f\small(0)\normalsize=0, f\small(1)\normalsize=2015. \\ (x-y)(f\small(f(x)^2\small)\normalsize-f\small(f(y)^2\small)\normalsize)=(f\small(x)\...
1
vote
2answers
90 views

Real functions with the property: $\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2 $ for all $\ x_1,\ x_2\in\mathbb{R}.\ $

Suppose $\ f:\mathbb{R}\to\mathbb{R}\ $ has the property:$\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2\ $ for all $\ x_1,\ x_2\in\mathbb{R}$. I made some educated guesses and stumbled upon the ...
-2
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1answer
63 views

If $f(f(x+y))= f(x^2) + f(y^2)$ then $f(2025) = ?$ [closed]

$f: \mathbb{Z} \to \mathbb{Z}$ $f(f(x+y))= f(x^2) + f(y^2)$ $f(f(2020)) = 1010$ Now: Find $f(2025)$ Find $f(x)$
1
vote
1answer
80 views

Let $f(x)$ be a fourth differentiable function such that $f(2x^2-1)=2xf(x)$, $\forall x \in \mathbb R$, then $f^{(4)}(0)$ is equal to?

Let $f(x)$ be a fourth differentiable function such that $f(2x^2-1)=2xf(x)$, $\forall x \in \mathbb R$, then $f^{(4)}(0)$ is equal to? Solution Given in text book: Replace $x$ by $(-x)$ in given ...
2
votes
2answers
122 views

if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$

if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$ for all integers ($f: \mathbb Z \rightarrow \mathbb Z$) I know how to solve the following problem though: if $f(f(x+y))=f(2x)+2f(y)$ then $f(x)=?$ We can ...
0
votes
0answers
27 views

Why is injectiveness needed to show that a bounded linear operator is a closed map?

Let $X$ and $Y$ be Banach. I'm trying to show that a bounded linear map $T:X\rightarrow Y$ is a closed map, that is $A\subset X$ is closed $\implies T(A)\subset Y$ is closed, if $T$ is injective and $...
2
votes
2answers
73 views

Is there any characterization for the real-valued function such that $y - x = y' - x' \Rightarrow f(y) - f(x) = f(y') - f(x')$

I'm looking for a real-valued function, $f: \mathbb{R} \rightarrow \mathbb{R}$, which is monotone, i.e., $x \leq y \Rightarrow f(x) \leq f(y)$, and with a specific characterization which can be called ...
2
votes
0answers
51 views

Solving Functional Equation from an Ordinary Generating Function

I was working with a friend on his CS homework, and one of his problems involved the following sequence: $$T(1) = 1; T(n) = T(n - 1) + T\left(\left\lfloor\frac{n}{2}\right\rfloor\right) \text{for } n &...
0
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0answers
37 views

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)\geq 0$, we have: $2f(-a)+f(b)\leq 0$. Then, $f$ must be an odd function?

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)=0$, we have: $2f(-a)+f(b)=0$. For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)>0$, we have: $2f(-a)+f(b)<0$. Also, for all $a$, there ...
0
votes
2answers
65 views

$f(a)+f(-b)=0$ implies $f(-a)+f(b)=0$. Then $f$ is an odd function?

If $f(a)+f(-b)=0$ for some $(a,b)$, then, $f(-a)+f(b)=0$. Also, for all $a$, there exists $b$ such that $f(a)+f(-b)=0$. $f:\mathbb R\to\mathbb R$ is continuous. Based on the above conditions, can we ...
3
votes
3answers
108 views

Find function $ f $ such that $f(\frac{x-3}{x+1})+f(\frac{x+3}{1-x})=x$

I am looking for functions $ f:\Bbb R \to \Bbb R $ satisfying $$f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{x+3}{1-x}\Big)=x$$ I used the substitution $ x=\cos(2t) $ for $ x\in (0,2\pi) $, to use the ...
1
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0answers
40 views

Find the smallest value at x=25

A function g: is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers m and n. Let f be an adjective function such that the value of $f(1)+f(2)+f(3)+...+f(30)$ is minimized. Find ...
0
votes
0answers
69 views

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$ My attempt: \begin{align} P(0, y): \; & f(f(0))=yf(0)=0. \\ \Rightarrow \; & f(0)=0. \\ \ \\ P(x, 0): \; & f(f(...
0
votes
2answers
78 views

Find all functions that satisfy $f(x^2f(y)^2)=f(x)^2f(y)$.

Find all $f:\mathbb Q_{>0}\to \mathbb Q_{>0}$ such that $$f(x^2f(y)^2)=f(x)^2f(y)$$ This is from the IMO Shortlist 2018, and I just want to know if my solution is valid, here is the official ...
1
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0answers
52 views

$\frac{g(x,y)}{f(x)-f(y)}$ is increasing with $x$ and decreasing with $y$. Then, what is $g$?

Background: $x,y,a$ are real numbers. $a>0$. $f(x), g(x,y), h(x,y)$ are continuous real functions. $g,h$ are defined on $[0,1]^2$. $$h(x,y)=\frac{g(x,y)}{f(x)-f(y)}>0, \text{when} \ x\neq y.$$ ...
4
votes
1answer
113 views

Find $f: \mathbb{N_0} \to \mathbb{N_0}$ which satisfies $f^n(x+f(y))=f^{n+1}(x)+f^n(y) \text{ for } n \in \mathbb{N}.$

Find $f: \mathbb{N}_0 \to \mathbb{N}_0$ which satisfies $$f^n(x+f(y))=f^{n+1}(x)+f^n(y) \quad \text{ for a given } n \in \mathbb{N}.$$ ($f^n$ means $\underbrace{f \circ f \circ f \circ \cdots \circ f}...
3
votes
1answer
151 views

Does $f(x+1)-f(x)$ constant imply $f$ is linear?

I have a function $f:\mathbb R\to \mathbb R$ and $f$ doesn’t have any restriction, so it might not be continuous and so on. If the condition $$f(x+1)-f(x)=c$$ holds $\forall x\in \mathbb R$, does that ...
2
votes
3answers
97 views

Is there any non-constant function $f(x)$ satisfying $f(x) f(y) = f(x) + f(y)$?

I am interested in the following functional equation: $\begin{equation} f \left(x \right) f \left(y \right) = f \left(x \right) + f \left(y \right) \end{equation}$ In particular, I would like to know ...
0
votes
1answer
84 views

Is my proof that $f(x)$ where $f(f(x)) = 6x - f(x)$ for all $f:R+→R+$ is linear correct?

The top functional equation was assigned in a Putnam competition. To prove that this function is linear, I did the following algebra: $$f(f(x))=6x-f(x)$$ $$f(f(x)) + f(x) = 6x$$ $$\frac{f(f(x))+f(x)}{...
3
votes
3answers
110 views

Find closed form of $f(x)= \frac{1}{x} \sum_{i=1}^{k} f(x+i)$

I'm trying to figure out the closed form of: $$f(x)= \frac{1}{x} \sum_{i=1}^{k} f(x+i)$$ $k$ is an integer greater or equal to 1. When $k=1$, the closed form is simply $f(x)=c_1\Gamma(x)$, where $c_1$ ...
2
votes
1answer
60 views

$f:\mathbb{R} \to \mathbb{R}$ such that $f(x+y^2+f(y)) = f(x-y^2-f(y))$

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x+y^2+f(y)) = f(x-y^2-f(y))$. There is actually a proof that $f\equiv c$ and $f(x) = -x^2$ are the only ones - it goes through the not too ...
0
votes
1answer
63 views

How to go about finding a function that satisfies $f(x)=-f(x-1)^{2}+2^{2^{x-3}}f(x-1)+2^{2^{x-2}}$ (Or determining if such a function exists)

I don't necessarily need an answer to this particular case, but in general I have no idea how to solve this kind of problem or even how to Google for such a method. If it helps for this particular ...
1
vote
0answers
75 views

$f: \mathbb{N} \to \mathbb{N}$, Find $f$ with Mathematical Induction of the value of $f(x)-f(x-1).$

$ f:\mathbb{N}\to\mathbb{N}. \\ \ \\\text{i)} \ \ \ \ f(x-1)f(x+1)=\big(f(x)\big)^2 \text{ for } \forall x \in \mathbb{N} \text{ s.t. } \frac x 2 \in \mathbb{N}. \\ \text{ii)} \; \; f(x-1)+f(x+1)=2f(x)...
0
votes
3answers
87 views

Find all functions s.t.$f(x^2-y^2)=xf(x)-yf(y)$ [duplicate]

Find all functions such that $\forall x,y\in \mathbb R$ $$f(x^2-y^2)=xf(x)-yf(y)$$ It’s obvious that $f(0)=0$, and by setting $x=0$ then $y=0$, we get $$\cases{f(x^2)=xf(x) \\ f(-y^2)=-yf(x)=-f(y^2)}$$...
-3
votes
1answer
142 views

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such as $x\left[f(x+y)-f(x-y)\right]=4yf(x)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such as $$x\left[f(x+y)-f(x-y)\right]=4yf(x)$$ My attempt: Let $x=0\Rightarrow 4yf(0)=0\Rightarrow f(0)=0.$ Let $y=x\Rightarrow xf(2x)=4xf(x)\Rightarrow ...
1
vote
0answers
73 views

Q : Let $f : \mathbb Z\to \mathbb Z$ such that, for all $x,y\in\mathbb Z:$ $f(f(x) − y) = f(y) − f(f(x)).$ Show $f$ is bounded.

I came to the conclusion that f is periodic with period |f(x0)| for x0 non-zero. But I don't see how the periodicity in domain translates to a bound in the codomain. It probably has something to do ...
-1
votes
0answers
31 views

Proving properties of the functional equation, $f(2n) = 2 f(n)$, and $nf(2n + 1) = (2n + 1)(f(n) + n)$ [duplicate]

The question is as follows and is from the BMO, The function f is defined on the set of positive integers by f(1) = 1, f(2 n) = 2 f(n), and nf(2 n + 1) = (2 n + 1)( f(n) + n) for all n ≥ 1. i) Prove ...
3
votes
1answer
66 views

Determine whether or not the function f it is bijective 2f(3-2x)+f(3/2-x/2)=x, where x is a real number

So the problem says: Study the bijectivity of the functions f:R->R, which follow the property: a) $2f(x)+f(1-x)=x-8$; b)$2f(3-2x)+f(\frac{3-x}{2})=x$, where x is a real number for both. The first ...
3
votes
1answer
119 views

Find all functions $f:\mathbb N\to \mathbb N$ such that $\frac{4f(x)f(y-3)}{f(x)f(y-2)+f(y)f(x-2)}$ is an integer for all $x>2$ and $y>3$.

The following problem is from an online contest that ended today: Find all functions $f:\mathbb N\to \mathbb N$ such that $$\frac{4f(x)f(y-3)}{f(x)f(y-2)+f(y)f(x-2)}$$ is an integer for all $x>2$ ...
1
vote
1answer
32 views

Maximising Property

Let $f:\mathbb{R}^2\mapsto\mathbb{R}$ and $g_1,g_2:\mathbb{R}\mapsto\mathbb{R}$ be continuous functions. Is it true that, for any $a_1,a_2\in\mathbb{R}$, $x$ is a maximiser of $$ f(g_1(x+a_1),g_2(x+...
0
votes
1answer
33 views

Sum of roots of a functional equation

Let $f(x)$ be a function which satisfies $f(29+x)=f(29-x)$ for all $x \in \mathbb{R} .$ Suppose $f(x)$ has (exactly) three real roots $a, b, c$, determine the value of $a+ b+c$. My work: From $$f(29 +...
1
vote
1answer
56 views

Find all $f: \mathbb{R} \to \mathbb{R}$ such that $(f(x)+y)(f(x-y)+1)=f(f(xf(x+1))-yf(y-1)), \forall x,y \in \mathbb{R}$

Find all $f: \mathbb{R} \to \mathbb{R}$ such that \begin{align}(f(x)+y)(f(x-y)+1)=f(f(xf(x+1))-yf(y-1)), \forall x,y \in \mathbb{R}\end{align} My attempts: I haven't been extremely good at $f$-in-$f$ ...
0
votes
0answers
53 views

Calculating the norm of a bounded linear functional in C[0,1]

Consider the space $C[0,1]$ equipped with the uniform norm. Let $$f(x)=\int_{0}^{\frac{1}{2}}x(t)dt-\int_{\frac{1}{2}}^{1}x(t)dt \qquad (x\in C[0,1]).$$ Now i need to calculate $||f||$.It's easy to ...
-1
votes
1answer
68 views

How to solve this functional equation $f(xf(y))+y+f(x)=f(x+f(y))+yf(x)$?

Find all function $f:\Bbb{R}→\Bbb{R}$ such that for all $x,y\in\Bbb{R}$ $$f(xf(y))+y+f(x)=f(x+f(y))+yf(x)~~~(1) $$ what I found is: if we plug $x=0$ we find $$2f(0)+y=f(f(y))+yf(0)~~~ (2)$$ so either $...
-3
votes
1answer
198 views

Functional equation in natural numbers with divisibility: $f(m) + f(n) + mn \ | \ m^2f(m) + n^2f(n) + f(m)f(n)$

Find all the functions $f : \mathbb{N}^* \to \mathbb{N}^*$ for which, for all $m, n \in \mathbb{N}^*$: $$f(m) + f(n) + mn \ | \ m^2f(m) + n^2f(n) + f(m)f(n)$$ Approach: For $m = n = 1$, the relation ...
0
votes
0answers
45 views

Showing these are the only functions in "Find All"

This question is not particularly difficult, in terms of finding the answers. I want to make sure the "proof" is complete, in particular the part of "find all" questions where you ...
2
votes
1answer
60 views

Decomposing an intensity spectrum as a superposition of blackbody spectra

My question goes as follows. The same way any integrable function $f(x)$ can be somewhat expressed as a superposition of plane waves as $\int_{-\infty}^{+\infty} F(\lambda)e^{2\pi i x\lambda}d\lambda$,...
-1
votes
1answer
106 views

Find all functions such that $f\left(x^2+y\right)=f(x)^2+\frac{f(xy)}{f(x)}$ in $\mathbb R^*$

Finds all function $f:\mathbb{R}^*\to\mathbb{R}^*$ such that $$\forall x,y\in\mathbb{R}^*,y\neq-x^2\qquad f\left(x^2+y\right)=f(x)^2+\frac{f(xy)}{f(x)}$$ where $\mathbb R^*=\mathbb R\setminus\{0\}$. ...
-1
votes
2answers
124 views

Determining all the functions $f:\mathbb{R}\to\mathbb{R}$ with the property $f\bigl(x-f(y)\bigr) f\bigl(f(x)+y\bigr)=x f(x) - y f(y)$ [closed]

Determine all the functions $f:\mathbb{R}\to\mathbb{R}$ with the property that $$f\bigl(x-f(y)\bigr) f\bigl(f(x)+y\bigr)=x f(x) - y f(y)$$ for any real numbers $x$ and $y$. I managed to see that $f(x)...
4
votes
1answer
140 views

Which functions share a certain property of sinusoids?

Among functions $f$ satisfying $\forall x\in\mathbb R\, f(x+p) = f(x)$ with $p>0$ are sinusoids $f(x) = A\sin(\omega x +\varphi)$ with $p=2\pi/\omega.$ These also satisfy the functional equations $$...
0
votes
1answer
65 views

The existence of bounded linear functional on a complex Hilbert space

Let $H$ be a complex Hilbert space and $\{e_n\}$ be an orthonormal basis. There exists a bounded linear functional $f:H \to \mathbb C$ such that $f(e_n)=\frac{1}{n}$. There exists a bounded linear ...
8
votes
0answers
194 views

Can every functional equation be solved or proved unsolvable?

Recently, I have read articles about how some identities whose "solution" either cannot be determined within $\mathsf{ZFC}$ or other axiomatic systems or the solvability is closely related ...
6
votes
2answers
190 views

Nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f\bigl(f(x)\bigr) = \frac{f(2x)}{2}$ [closed]

I'm looking for a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at no point and satisfies the identity $$f\bigl(f(x)\bigr) = \frac{f(2x)}{2}$$ for all $x \in \mathbb{R}$. This is not a ...
1
vote
1answer
119 views

Finding all polynomials $P(x) \in \mathbb R[x]$ such that $P(x)^2=4P\left(x^2-5x+1\right)+2$

Find all polynomials $P(x) \in \mathbb R[x]$ such that $P(x)^2=4P\left(x^2-5x+1\right)+2$. This comes from a no-solution class problem so it should have a definitive solution, unless my teacher ...
4
votes
1answer
226 views

Solution to the functional equation $f(2x) = f(x)\cdot\sin(x)$?

Solution to the functional equation $f(2x) = f(x)\cdot\sin(x)$ ? At first I believe that finding an answer to that equation it was going to be an easy problem, since this other equation $g(2x) = g(x)\...
0
votes
1answer
181 views

If $\frac{f(x^2)}{f(x)}=1+x+x^2+\ldots+x^7$ then what is $f(x)$?

I found this problem on an Instagram page which send mostly challenging problems. $$\frac{f(x^2)}{f(x)}=1+x+x^2+\ldots+x^7\qquad\qquad f(x)=?$$ For $x=1$ we get $1=8$ which is a contraction hence $x=...
0
votes
1answer
78 views

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(2x^2+2yf(z))=2xf(x)+2zf(y).$

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(2x^2+2yf(z))=2xf(x)+2zf(y).$ My attempt: \begin{align} & \text{if } f \equiv 0: \text{Solution.} \\ & \text{if } f \not\equiv 0: \\ \ \\ ...
-1
votes
1answer
96 views

What is the derivative of $f(x)$ if $f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$?

Question. What is the derivative of $f(x)$ if $f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$? So my solution is the following: Differentiating with respect to $x$ gives $$ f'(x) = f '\left(\frac{x+y}{1-...
2
votes
3answers
118 views

Find all $f(x)$ such that $x(f(x+1)-f(x))=f(x)$

The problem Find all $f(x): \mathbb{R} \to \mathbb{R} $ such that $x(f(x+1)-f(x))=f(x)$ and $|f(x) -f(y)| \le |x-y|, \forall x,y \in \mathbb{R}$ My approach Obviously $f(x)=x$ is one solution, I ...

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