0
votes
3answers
36 views

Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
3
votes
2answers
50 views

Find all functions that satisfy the following conditions

Find all functions $f:\mathbb Z\to \mathbb Z$ that satisfy the following conditions: (i) $f (0) = 1 $ (ii) $f(f (x)) = x$ for all integers x (iii) $f(f(x + 1)+1) = x$ for all integers x How ...
7
votes
3answers
235 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
1
vote
1answer
25 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
10
votes
1answer
121 views

What is the solution of the following functional equation? (I must confess it is a headache for me)

Find all the functions $f: \mathbb{Z} \to \mathbb{Q} $ such that $f(\frac{x+y}{3})=\frac{f(x)+f(y)}{2}$; $\forall x,y\in\mathbb{Z}$ knowing that $\frac{x+y}{3}\in\mathbb{Z}$.
7
votes
1answer
203 views

Find all continuous functions $ f:\mathbb{R} \rightarrow \mathbb{R}$ such that

Find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x)^2 - 2\Bigl\lfloor\,2\,\bigl|f(2x)\bigr|\,\Bigr\rfloor f(x) + \Bigl\lfloor{4f(2x)^2-1}\Bigr\rfloor = 0\,.$$ Here ...
0
votes
1answer
34 views

functional equation, find $h$ continous on $\mathbf{R}$ such that $h(x) + h(2x) + h(4x) = {x^n}$

I am having trouble finding easily $h$ defined and continuous on $\mathbf{R}$ verifying for all $x$ in $\mathbf{R}$, $$ h(x) + h(2x) + h(4x) = {x^n} $$ where $n$ is a fixed natural number. I have a ...
9
votes
2answers
257 views

How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
0
votes
1answer
358 views

fabric design using trigonometric functions

is there any trigonometric function or any others that involve trigonometric function, that draw cool fabric shapes or patterns? I have seen some pictures like but with trigonometric functions... ...
-2
votes
2answers
83 views

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$ [duplicate]

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$.
3
votes
2answers
75 views

Finding a function from the given functional equation .

The question asks us to find the function $f(x)$ with the given information Let $f:\mathbb R \rightarrow \mathbb R$ such that $f'(0)=1$ and $f(x+y)=f(x) + f(y) + (e^{x+y})(x+y)-xe^x-ye^y+2xy$ ...
3
votes
1answer
153 views

Find all polynomials $P(x)$ satisfying this functional equation

Find all polynomials $P(x)$ which have the property $$P[F(x)]=F[P(x)], \quad P(0) = 0$$ where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$. This is an exercise ...
4
votes
2answers
119 views

Prove that if a particular function is measurable, then its image is a rect line

I´m really stuck with this problem of my homework. I don´t have any idea, how to begin. Let $f$ be a function, $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$ , $\forall ...
7
votes
1answer
265 views

How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?

Can someone please show me how to: Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$? I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
13
votes
4answers
530 views

Polynomial: $p(x) = p(x+3)$.

Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$. Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
5
votes
1answer
185 views

$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in \mathbb{Z^+}$

Find all the function $f:\mathbb{R}\rightarrow \mathbb{R}$ sastisfied that $f$ continuous on $\mathbb{R}$ and $$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 ...
1
vote
1answer
41 views

Funcional Equations:I'm confused [duplicate]

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$ Or : ...
0
votes
2answers
60 views

What's the solution of the functional equation

I need help with this: "Find all functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, with $g$ injective and such that: $$f(g(x)+y) = g(f(x)+y), \mbox{ for all } x, y \in \mathbb{Z}.$$
1
vote
1answer
61 views

What's the solution of the functional equation?

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
3
votes
2answers
569 views

How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?

This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that ...
7
votes
3answers
412 views

Find all the functions which satisfy a given functional equation

I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$ We ...
1
vote
1answer
34 views

Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $

Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$. Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected ...
3
votes
2answers
92 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 ( ...
3
votes
2answers
578 views

Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$

I've been working on the following homework problem: Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$. The first ...
6
votes
2answers
290 views

Proving a function is constant when $f(x)f(y) + f(\frac{a}{x})f(\frac{a}{y}) = 2f(xy)$

I've been working on the following homework problem: Consider a function $f : (0,∞) → \mathbb{R}$ and a real number $a > 0$ such that $f(a) = 1$. Prove that if $f(x)f(y) + ...
0
votes
0answers
32 views

Proving Join Dependencies in MVD

I have a question regarding natural join operations in multivalued dependencies. I know that a join operator joins two tables on similair attributes, however I have a hard time to figure out how to ...
-2
votes
2answers
170 views

Find all functions : $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that:

Find all functions : $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that: $$f(1+xf(y))=yf(x+y)$$ for all $x,y \in \mathbb{R^+}$
3
votes
2answers
166 views

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that:

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that: $$f(x)\cdot f(yf(x))=f(y+f(x))$$ $\forall x,y \in \mathbb{R}^+$
1
vote
0answers
102 views

Periodicity of a function.

If we have two functions $f:\mathbb{R} \to \mathbb{R}$ and $g :\mathbb{R} \to \mathbb{R}$ such that the period of $f$ is 7 and that of $g$ is 11, then the period of $F\left ( x \right ) = f( ...
3
votes
1answer
1k views

Solve the recurrence for $T(n) = T(\sqrt n) + 2$. Assume that T(n) is constant for $n\leqslant 2$.

Trying to work out the following question, but I'm stuck.. Can someone direct me please. Using a change of variables $$ \text{Let}\ m = \lg\ n \\ S (m) = T (2m)\\ T (2^{m}) = T ...
5
votes
2answers
84 views

$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$

Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...
2
votes
2answers
147 views

Find $f(x)$ from $f(f(x))$

I have this: $$ f\colon \mathbb{R} \to \mathbb{R}, $$ $$ f(f(x)) = x^2 - x + 1 $$ I need to show that $f(1) = 1$ and I need to show that $g(x) = x^2 - xf(x) + 1$ is not an one-to-one fuction. I know ...
0
votes
1answer
209 views

find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$ Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a<b, \exists c \in (a,b): ...
5
votes
1answer
128 views

Functional Equation with Value

If $f$ is a strictly increasing function from the naturals to the naturals, and $f(f(x))=3x$, what are all values of $f(2012)$? I have only proven that $f(3x)=3f(x)$ but that get's nowhere :(
1
vote
1answer
475 views

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Possible Duplicate: Proving that an additive function $f$ is continuous if it is continuous at a single point Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) ...
0
votes
1answer
85 views

General solution to homogeneous difference equation

With a given example $$ a_{n-1} = ca_{n-2} $$ general solution: $$ a_{n} = c . c . a_{n-2} $$ $$ = c . c . a_{n-3} $$ $$ = c^n a_0 $$ Question: Find the general solution for the ...
0
votes
3answers
180 views

Recurrence relation - How to solve this recurrence relation

a person invests 1000 at a bank at 4 percent compound interest compounded annually and every year government and bank charges amounting to C are deducted and if An is the value of the investment at ...
0
votes
1answer
158 views

Solving general solution of recurrence relation by iteration

$$a_{n-1} = ca_{n-2} $$ Hence $$a_n = c \cdot c \cdot a_{n-2} $$ $$ = c \cdot c \cdot c \cdot a_{n-3} $$ ...... $$ = c^na_0 $$ Why is there a iteration on the constant $c$ ?
3
votes
2answers
212 views

Understanding difference equation

I was given an example $$R_n = R_{n-1} + R_{n-2} $$ This equation is given as an second-order equation. Why is it so?