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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Functional equation $f(h(y)x+y)=g(y)f(x)$

(Note: this is a simplified version of my previous question, which was not answered). I am seeking the solution for the functional equation $f(h(y)x+y)=g(y)f(x)$ where $f,g,h$ are continuous. ...
1
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2answers
131 views

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^{2}$ . Then $f(3)$ =? [closed]

Options: (a)$4$, (b)$4f(0)$, (c)$4-f(0)$, (d)$4+f(0)$, (e)$16+f(0)$. CORRECT ANSWER USING REDUCTION Deep thanks to @martini and @A.S. , soo ...
5
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1answer
80 views

Solving functional equation $2f(x) = f(2x)$

$f(x)$ is a $\mathbb{R} \rightarrow \mathbb{R}$ differentiable function satisfying the following equation: $$2f(x) = f(2x).$$ Can it be proved that $f(x) = kx$ for some $k$? Note that if $f(x)$ is ...
3
votes
2answers
62 views

Functional equation $f(x+y)=f(x)+2xy+f(y)$

I am interested in classifying solutions $f\,:\,\mathbb R\longrightarrow \mathbb R$ to the functional equation \begin{equation} f(x+y)=f(x)+f(y)+2xy\qquad\qquad(\dagger) \end{equation} and in ...
2
votes
3answers
77 views

What is the vertex of this radical equation?

the question is $$y = \sqrt{-(x -3}) + 4$$ i thought the vertex was (3,4) but i was wrong and that it was supposed to be (3,2). Was i right, can anyone help me with these type of question?
1
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1answer
52 views

Solving functional equation $f:Q^+\to R^+$ where $f(xy)=f(x+y)(f(x)+f(y))$

Find all functions $f:\mathbb{Q}^+ \to \mathbb{R}^+$ with the property: $$f(xy)=f(x+y)(f(x)+f(y)),\qquad \forall x, y\in\mathbb{Q}^+ \tag{1}$$ This question is from the 2014 Bulgaria National ...
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0answers
47 views

Cauchy-like functional equation $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on ...
5
votes
2answers
73 views

Finding a function (?) and computing its definite integral

So I've come across this exercise from one of my old highschool textbooks: $$\text{If}\ 2f\bigg(\frac{x-2}{x+1}\bigg) +f\bigg(\frac{x+1}{x-2}\bigg) = x$$ Considering this, find : ...
1
vote
1answer
27 views

Any chances this can be further reduced?

I've come with the following equation, after a lot of simplification, but can't reduce further. Any chances it can be solved by reducing the $b$ and get the value of $a$? $$a = \frac{1000(1000 - ...
4
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2answers
120 views

Prove that $f′(x)=f′(0)f(x)$ derivatives

Let $f:I \to R$ be differrentiable on an open interval $I \subseteq R$ with $$f(a + b) = f(a)f(b) \quad \forall a, b \in R$$ Suppose that $f(0) = 1$ and that $f'(0)$ exists. Show that: $$f'(x) = ...
0
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1answer
88 views

A nice form of a given function

First let, $\oplus(a_1,a_2,\ldots,a_n)$ denote the bitwise xor of $a_1,a_2,\ldots,a_n$. Define the function $\Delta(a_1,a_2,\ldots,a_n)$ to be the maximum value of $a_i - ...
3
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1answer
96 views

Find continuous functions that satisfy $f(f(x))=x$ over the reals.

I'm looking for a method to solve: $$f(f(x))=x$$ Where $f$ is defined for $x \in R$ So far by inverting both sides I have: $f(x)=f^{-1}(x)$ Which means that my function should be symmetrical over ...
3
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1answer
74 views

How to solve this nonlinear functional recurrence

I study two similar nonlinear functional recurrence systems, given by $$P_\pm:\qquad f_n\cdot(1\pm g f_{n-1}) = g\mp(1+2g)f_{n-1} \qquad (n>0)$$ and $$f_0=g$$ Here $f_n$ and $g$ are functions of ...
2
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1answer
39 views

How to algebraically prove the following inequation?

Following is the inequation I have been trying to prove for a while. $$\frac{\frac{1}{2}(1-q)}{\frac{1}{2}(1-q) + pq}\neq \frac{\frac{1}{4}(1-q)}{\frac{1}{4}(1-q) + p^2q} + ...
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2answers
36 views

Obtaining values from functional equation without solving [closed]

Let $$f(x)=\frac{1}{2}[f(xy)+f(x/y)]$$ for real positive x,y such that f(1)=0 and f'(1)=2. How to find f'(3) and f(e) without explicitly solving the recursion?Any suggestions?
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1answer
99 views

Find $f:\mathbb{R}\to\mathbb{R}$ such that $f(xy+x+y)=f(xy)+f(x)+f(y)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function that satisfies $$ f(xy+x+y)=f(xy)+f(x)+f(y) $$ Find $f$ and prove that $$ f(x+y)=f(x)+f(y) $$
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2answers
32 views

How to rewrite $x-y=\frac{x}{y}$ so that it become $y=$ (something…)?

For example, $x+y=x\times y$ is easy to express as $y=\frac{x}{x-1}$, how about $x-y=\frac{x}{y}$? I tried multiply both sides by $y$ and become $y^2-xy+x=0$ but up to this step I don't know how to ...
5
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3answers
310 views

Functional equation $f(x+y)+f(x-y)=2f(x)f(y)$

Let $f:\mathbb{R}\to\mathbb{R^*}$ be a function such that $f(x+y)+f(x-y)=2f(x)f(y),\forall x,y\in\mathbb{R}$. Prove that $f(x)=1,\forall x\in\mathbb{R}$. I have managed to prove the following: 1) ...
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0answers
33 views

Finding Linear Operator for a given Basis

Consider a linear operator $$L: \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace \rightarrow \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace $$ For example $$ L(f) = f(x+1) - f(x)$$ Define the ...
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1answer
29 views

Find a 3d equation that goes through a series of points?

I have a series of points in 3d space and I need to find an equation that goes through all of them. What would be the best way to do this? Points: (3.7, 0.45, 0.7) (5.2, 0.8, 0.96) (6, 1.04, 1.15) ...
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1answer
34 views

How to find a 3d equation from a series of points

I have 6 points and I need to find the equation, or an equation, that will go through all of them. How would I go about doing this? The points are as follows. (3.7, 0.45, 0.7) (5.2, 0.8, 0.96) (6, ...
6
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0answers
100 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
1
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2answers
97 views

How can I solve this recursive function $f(n) = f(f(n+1))$?

I am trying to solve this: $$ f(n) = \begin{cases} n - 1,& n > 5\\ f(f(n+1)),& n\leqslant 5 \end{cases} $$ What is the technical name of this kind of function ? --> ...
4
votes
2answers
66 views

How to solve $f\left(\frac{f(x)}{yf(x)+1}\right)=\frac{x}{xf(y)+1}$?

I'm currently working on the following functional equation: Find all $f:\mathbb{R_{>0}}\to\mathbb{R_{>0}}$ such that for all $x,y\in\mathbb{R_{>0}}$: $$ ...
0
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2answers
82 views

Find a function $f$ which satisfies $f(mn) = f(m)f(n)$ for positive integers $m,n$ and $f(2)=2$

We are to find a function f which follows the following properties $$f(mn)=f(m)f(n),\; f(2)=2.$$ I can easily find all the values of $f(2^n)$ but I am confused on how to find for the odd numbers and ...
1
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1answer
34 views

Why does the Lagrange equation have to be zero?

I know it's a pretty basic question, but I still don't get it since starting Lagranian mechanics this year. I tried to read Stone and Goldbart's "Mathematics for Physics" and they said: Suppose ...
3
votes
1answer
74 views

$f(x) \ge f(x + \sin x)$, nonconstant functions, infinite number of solutions to $f'(x) = 0$.

Let $\mathcal{F}$ be the set of all the differentiable functions $f: \mathbb{R} \to \mathbb{R}$, which have the property $f(x) \ge f(x + \sin x)$, for all $x \in \mathbb{R}$. Prove that ...
7
votes
2answers
77 views

How to solve $\frac{f^{-1}(x)f(x)}{x}=\frac{f^{-1}(x)+f(x)}{2}$?

I've come across the following functional equation: Determine all surjective functions $f:\mathbb{R_{>0}}\to\mathbb{R_{>0}}$ which satisfy for all $x\in\mathbb{R_{>0}}$: $$ ...
1
vote
1answer
87 views

Find all real functions $f$ such that $f(xf(y))=(1-y)f(xy)+x^2y^2f(y)$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$ we have: $f(xf(y))=(1-y)f(xy)+x^2y^2f(y)$ I have solved this problem but the solution is ...
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0answers
43 views

What is the solution of this recursion, that's defined in terms of a sum, but with this $1$ odd twist?

$$ F(n) = \sum_{i=0}^{n} F\left(\left\lfloor\frac{i}{5}\right\rfloor\right) $$ I encountered this odd looking functional equation, while perusing the site yesterday. I'd be interested in seeing a ...
4
votes
1answer
70 views

How to solve $f(x+f(x)+2f(y))=f(2x)+f(2y)$?

Another functional equation: Find all surjective functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$ it satisfies: $$ f(x+f(x)+2f(y))=f(2x)+f(2y) $$ I couldn't make any progress ...
3
votes
1answer
68 views

Is there any neat way to show $T$ is $ \mathbb R$-linear?

Let $T: \mathbb R \to \mathbb R$ be the map which satisfies the following functional equation $T(x^2+T(y))=y+T(x)^2$ $ \forall x,y \in \mathbb R$ Is there any neat way to show that $T$ is $ \mathbb ...
7
votes
2answers
121 views

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

How to solve the following functional equation: Find all $f:\mathbb{R}\to\mathbb{R} $ such that: $$ f\left(x^2+f(y)\right)=y+f(x)^2 $$ Holds for every $x,y\in\mathbb{R}$. A friend gave it to me, ...
0
votes
1answer
59 views

A simple two variable functional equation

A real function $f(x,y)$ on $R^2$ satisfies $f(x+y,y) = f(x,0)+qy$ for some real number $q$. What form should $f$ assume without $f$ being continuous? Is the linear solution $f(x,y)=ax+(q-a)y$ for ...
11
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1answer
240 views

Solve $f(x)f(y)=2f(x+yf(x))$

I want to solve the following functional equation: Find all functions $f:\mathbb{R_{>0}}\to\mathbb{R_{>0}}$ for which $$ f(x)f(y)=2f(x+yf(x)) $$ For all $x,y\in\mathbb{R_{>0}}$. It is from ...
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2answers
26 views

Find the value of K in a specific case of a cartesian plane

I have a linear equation of a line in a cartesian plane $r:= \{(x,y) \in \mathbb{R}^2 \mid kx-(k+1)y+k-1=0, \,\, k \in \mathbb{R}\}$ and I have to find the value of k so that the line intersects the ...
0
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1answer
20 views

Equation with variable $λ$ that should be $R$ for $x$

I need to find for $λ\in R$ the domain of $f(x)=\sqrt{((λ-2)x^2-2λx+2λ-3)}$ It should be $λ\in[6,+∞ )$ as per my book but I dont understand why. Sorry for my english
3
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1answer
55 views

Solving Functional Equations by Limits …

Let $f(x)$ be a continuous function and satisfying the equation : $f(2x) - f(x) = x$. Given $f(0)=1$ ; Find $f(3)=?$ My teacher solves this as : $$f(x) - f(x/2) = x/2$$ $$f(x/2) - f(x/4) = x/4$$ ...
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1answer
27 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
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1answer
38 views

Given a functional equation, its nature, value of its differential at a critical point, what are some methods to calculate its integral over a period?

My particular question is : If $f$ be a decreasing continuous function satisfying $$f(x+y)=f(x)+f(y)-f(x)f(y)$$$$ \forall x,y \in \mathbb{R}; f'(0)=-1$$ then $$\int_0^1f(x)dx =?$$ Answer to this ...
3
votes
1answer
93 views

What are solutions of that functional equation?

How to find all the functions $f:[0,\infty)\rightarrow [0,\infty)$ satisfying the functional equation $$ f(f(x))=-4f(x)+3x?$$ I deduced $f(x) \le \frac 3 4 x.$
3
votes
1answer
53 views

Functional equation: $f(f(x))=k$

If $k\in\Bbb R$ is fixed, find all $f:\Bbb R\to\Bbb R$ that satisfy $f(f(x))=k$ for all real $x$. If $k\ge 0$, $f(x)=|k+g(x)-g(|x|)|$ is a solution for any $g:\Bbb R\to\Bbb R$.
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vote
1answer
36 views

Functional equations with involutions

Having seen that the topic of functional equations is loved by StackExchange, I have constructed this problem hoping that it will please readers. Solve the functional equation $$ ...
2
votes
1answer
61 views

Sufficient conditions for this function being linear [duplicate]

Let $f$ be a real-valued function for which, for every real $x,y$: $$f(x+y) = f(x)+f(y)$$ Does this imply that $f$ is a linear function ($f(x)=a\cdot x$)? If $f$ is differentiable, I think the ...
6
votes
1answer
212 views

Transformation of the functional equation $f(x+y)=f(x+1)f(y)$

Is there a way to reduce the following functional equation $$ f(x+y)=f(x+1)f(y),\qquad x,y>0, $$ to the equation $$ f(x+y)=f(x)f(y),\qquad x,y>0, $$ whose solutions are known? Thanks in ...
1
vote
1answer
59 views

A problem on analysis specifically on functions

Let $f(x)$ be a function from reals to reals obeying the following: $f(x)$ is continuous, $f(0)=1$, and $f(m+n+1)=f(m)+f(n)$. Show that $f(x) =1 +x$ for all real numbers $x$. I am a bit confused on ...
1
vote
1answer
35 views

Understanding of solution for a functional equation.

Problem For all $x,y \in \mathbb{R}$ which is $x^2 \not = y^2$, a function $f$ satisfies the following. $$(x-y)f(x+y) - (x+y)f(x-y) = 4xy(x^2 -y^2)$$ Find the function $f$. Solution Divide ...
11
votes
4answers
334 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting ...
2
votes
3answers
81 views

Prove the equation has unique class of solutions

Find the solutions of equation: $$ x^y + y^x = 1 + xy \quad x,y \in \mathbb{R} \quad x,y >0 $$ My quest First, $x=1$ or $y=1$ gives us obvious solutions, so let's suppose $x \not =1$ and $y \not= ...
1
vote
2answers
48 views

Expressing Math Equations

I'm confused how to express the following expressions in math equations for publication: $x =$ integer part of $y$ $x =$ fraction part of $y$ image $x =$ shifted version of image $y$ left with $z$ ...