Tagged Questions
0
votes
0answers
19 views
Formulating math problem with rounding / discrete step
I have this problem, that can easily be solved by simulation or numerical optimization, but I wonder how to write it as a mathematical problem? It's two pricing schemes, one cost is evaluated at ...
6
votes
2answers
116 views
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
5
votes
2answers
64 views
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that
$$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$
I've tried subbing in heaps of values but I keep getting things like ...
2
votes
1answer
85 views
Functional equation $f(y/x)=xf(y)-yf(x)$
Are there any solutions to $f:\mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}$ with $f(\frac{y}{x})=x \cdot f(y)-y \cdot f(x)$ other than $f(x)=0$ for all $x \in \mathbb{R}$?
I already have found ...
10
votes
1answer
318 views
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
5
votes
2answers
133 views
$f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$
How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$
($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
1
vote
1answer
64 views
Injectivity of a Function [closed]
Sorry for confusion.
I am in the process of solving a functional equation, I need to show injectivity. (By the way i know that it is injective, I'm trying to prove it to myself).
Putting $f(x)=f(y)$ ...
3
votes
1answer
58 views
Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)
If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
11
votes
4answers
310 views
$f(16x)=16f(x) $ and $ f$ is continuous
$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$.
Should it be $f(x)=ax$? How can I prove that?
5
votes
1answer
120 views
Does a function exist with the property $f(-n^2+3n+1)=(f(n))^2+1$?
Let $f:\mathbb{R}\to \mathbb{R}$ be a function which fulfills for every $n \in \mathbb{N}$
$$f(-n^2+3n+1)=(f(n))^2+1$$
Is it possible that such a function exists?
6
votes
1answer
75 views
How can find this function by $x\in \mathbb{Q}^+$
Let $f:\mathbb{Q}^+\to \mathbb{Q}^+,f(x)+f(1/x)=1 $ and $f(2x)=2f(f(x)),x\in \mathbb{Q}^+$, prove that
$$f(x)=\dfrac{x}{x+1},x\in \mathbb{Q}^+$$
This Problem from my student.
2
votes
1answer
52 views
Connecting functional identity of a function with its image set
Okay, now I really need real help, maybe the task is not too heavy but I do not know the easy way to solve this. Let´s start with the problem, now.
Suppose that we have some function $f: \mathbb N ...
3
votes
2answers
80 views
Average of function, function of average
I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$:
$$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( ...
2
votes
2answers
82 views
Find the equation of this bijection from $ \mathbb{R} $ to $ (0,1) $.
I need some help (hints or an answer) in finding the actual equation of this bijections from the reals $ \mathbb{R} $ to $ (0,1) $. We may assume that the radius of the circle is $ \dfrac{1}{2} $.
...
0
votes
0answers
28 views
Scrambling an Array and then Recovering It
I have an array of elements which is finitely long -- typically < 2000. Each element can be between 0 - 0xFF
An example:
[0xa 0x1 0x7 .... 0x1f]
Are there any functions available to scramble ...
-6
votes
2answers
84 views
continuity and fixed point
Show that a continuous mapping f from $[0,1]$ to $[0,1]$ which satisfies $f(f(x)) = x$ for all $x\in [0,1]$ and for which $f(x) \neq x$ for at least one $x\in [0,1]$ must have exactly one fixed point.
...
10
votes
2answers
250 views
When is $f^{-1}=1/f\,$?
I seem to remember a nice article that I read many years ago (perhaps in the American Mathematical Monthly) which investigated the question, "Under what conditions on $f:\mathbb{C}\to\mathbb{C}$ is ...
-1
votes
1answer
47 views
Exercise of functions of a real variable
Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...
1
vote
1answer
70 views
Given $2xf(x)+(x-3)f(\frac{1}{1-x})=4x^2-10x-\frac{1}{2}$, find $f(x)$.
Given$$2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}$$ Find $f(x)$.
This's the first time I see this kind of question, I have no idea. Please help. Thank you.
4
votes
2answers
99 views
Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that :
Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that :
$$f(1)=1$$
$$f(x+y)=f(x)+f(y)+2xy$$
$$f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$$
0
votes
3answers
48 views
Prove: we always have at least x>0 is a $x^3+bx^2+cx-d^2=0$ 's root
Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)
0
votes
3answers
123 views
Find all function $f$ such that $f(x)+f(\frac1x)=\frac1a; a$ is constant
Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?
8
votes
1answer
55 views
Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?
This is my first question and I hope this question is not too brief to be acceptable:
There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
16
votes
4answers
638 views
Given $f(f(x))$ can we find $f(x)$?
Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
0
votes
1answer
102 views
Find all positive functions of a positive real such that $ f(xf(y))=yf(x) $ and $\lim_{x\to\infty}f(x)=0$
Find all functions defined on the set of positive reals which take positive real values and satisfy:
$$ f(xf(y))=yf(x) $$
for all ; $ f(x)\to0 $ and as $ x\to\infty $
1
vote
2answers
72 views
Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties
Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that:
$$f(x)f \left(\frac{1}{x}\right)=1$$
3
votes
3answers
96 views
Find the value of f(343, 56)?
I have got a problem and I am unable to think how to proceed.
$a$ and $b$ are natural numbers. Let $f(a, b)$ be the number of cells that the line joining $(a, b)$ to $(0, 0)$ cuts in the region $0 ≤ ...
37
votes
2answers
537 views
Looking for a function such that…
There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is:
...
1
vote
1answer
56 views
inverse function and positivity
is there any proof or theorem to say that if the inverse function $ y=f^{-1}(x) $ is POSITIVE in the sense $ f^{-1}(x) >0 $ for $x \ge 0 $ then the function $ f(x) \ge 0 $ will be also positive on ...
5
votes
3answers
273 views
Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$
I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given:
$$y=W(e^{ax+b})-W(e^{cx+d})+zx$$
where $W$ is the Lambert $W$ function and ...
1
vote
0answers
55 views
Functional equation for the given function
For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$
And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$.
At the same time, there is no known functional ...
2
votes
1answer
82 views
Symmetric homogeneous functions of degree 1
Suppose:
$cf(x,y)=f(cx,cy)$
$f(x,y)=f(y,x)$
If $f$ is a polynomial, then $f(x,y)=c(x+y)$ because by Euler's homogeneous function theorem, $f(x,y)=xf_x(x,y)+yf_y(x,y)$ where $f_x,f_y$ are ...
5
votes
2answers
215 views
Are all multiplicative functions additive?
Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$.
Proof: There is some $c$ such that $y=cx$. Then
...
11
votes
2answers
138 views
About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$
On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
1
vote
5answers
152 views
$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$. For what value of $p$, $g(x+p)=g(x)$.
$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$.For what value of $p$, $g(x+p)=g(x)$.
$g(x+2)+g(x)=g(x+1)$
1
vote
1answer
72 views
Show that for this function the stated is true.
For the function
$$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$
show that
$$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$
Hey everyone, I'm very new to this kind of maths and would really ...
3
votes
1answer
131 views
What is this topic (competition questions)?
I discovered the following question by accident, and found it interesting, but I only resolved it by brute force:
Problem: Let f: $\mathbb{R} \rightarrow \mathbb{R}$ have the property $(\pi)$ iff ...
3
votes
2answers
152 views
Does anyone recognize this function?
I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall ...
2
votes
2answers
107 views
Positive twice differential decreasing function, is it convex?
If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
1
vote
1answer
379 views
continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.
$g(0)=1$. If $a=g(1)$,then $a>0$ ...
2
votes
1answer
212 views
About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies
$$f(xy)=f(x)f(y)-f(x+y)+1$$
for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.)
How to show that ...
7
votes
7answers
350 views
Function satisfying $x = f(f(x))$ and $x \not= f(x)$
Is there a function that would satisfy the following conditions?:
$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,
where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
1
vote
1answer
165 views
Any example of functions are automorphism?
I am looking for functions fulfilling
$f(x+y) = f(x) + f(y)$ and $f(x*y) = f(x)*f(y)$.
I can only find $f(x)=x$, any more?
Any example of functions are automorphism?
5
votes
1answer
80 views
Finding a real value of $p$
I am a bit confused about approaching this problem,
Let $g(x)$ be a function such that $g(x +
1) + g(x − 1) = g(x)$ for every real $x$.
Then for what value of p is the
relation $g(x + p) = ...
11
votes
2answers
175 views
Can every real function be represented as two shifted even functions?
I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
2
votes
3answers
85 views
Defining a function with certain properties
I'm a bit rusty in mathematics so I need your help please :)
I need a function $y$ that satisfies:
$$\begin{align*}
y &= ax\\
y &= \left\{\begin{array}{ll}
x &\text{if }x\geq 0;\\
0 ...
5
votes
3answers
134 views
Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$
I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas.
PS. I've tried plotting the the initial values into ...
0
votes
2answers
38 views
Reversing bijections defined via conditional expressions
Let's say that I have a variable $j$ defined by the following formula:
$$j=\frac{n(n+2) + m}{2}$$
where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions:
$n\in ...
7
votes
4answers
494 views
Solving the functional equation $f(x+1) - f(x-1) = g(x)$
Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies
$$ f(x+1) - f(x-1) = g(x) $$
3
votes
0answers
73 views
Function Shape Reference
I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague.
I would like to have a ...
