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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5
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2answers
123 views

Is there a function satisfying the following properties $ f^{n}(x)=(f(x))^{n+1}$??

Is there a function with the following properties? $$ f(x)=f(x) $$ $$ f'(x)=f(x)^2 $$ $$f^{(n)}(x)=\left(f(x)\right)^{n+1}$$ where $f^{(n)}$ denotes the $n$th derivative, and by convention ...
3
votes
3answers
169 views

if I know $f(x+1) = 2f(x) + 1$, how do I solve f(x)

This is just my thought on run time of a binary search: if you are allowed to make 1 comparison, you can search a sorted list of length 1, but if you are allowed to perform 2 comparisons, you can ...
0
votes
2answers
51 views

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
vote
2answers
130 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
votes
1answer
72 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
56 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
52 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
vote
1answer
43 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 ...
0
votes
0answers
44 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
70 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 : ...
0
votes
1answer
25 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
votes
2answers
117 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
votes
1answer
86 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
votes
1answer
89 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
votes
2answers
73 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
votes
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} + ...
-1
votes
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?
2
votes
1answer
98 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 $$ f(x+y)=f(x)+f(y) $$
0
votes
2answers
31 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
votes
3answers
208 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) ...
1
vote
0answers
32 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 ...
0
votes
1answer
26 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) ...
0
votes
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, ...
4
votes
0answers
89 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
vote
2answers
96 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
64 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
votes
2answers
76 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
vote
1answer
32 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
72 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
73 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 ...
1
vote
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
69 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
66 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
111 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
46 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 ...
10
votes
1answer
222 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 ...
1
vote
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
votes
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
votes
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 : $$f(x) - f(x/2) = x/2$$ $$f(x/2) - f(x/4) = x/4$$ ...
1
vote
1answer
26 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 $y(x)=Cx^4$.
0
votes
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
91 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
52 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 $g:\Bbb R\to\Bbb R$.
1
vote
1answer
34 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
58 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
211 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
58 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
34 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 ...
10
votes
4answers
268 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 ...