5
votes
2answers
152 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
1
vote
1answer
26 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
8
votes
2answers
228 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
0
votes
1answer
34 views

Finding equations when given new center of a circle

$y = −x + \sqrt{2}$, $y = −x − \sqrt{2}$, $y = x + \sqrt{2}$, and $y = x − \sqrt{2}$. These equations determine lines, which in turn bound a diamond shaped region in the plane. Construct a diamond ...
3
votes
2answers
59 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
2
votes
2answers
63 views

How do you find two functions $f$ and $g$ such that $f(x) \cdot g(x)=f(x)-g(x)$?

This was inspired by this question ( Logarithms with trigonometric inequality ) I already know the answer ( $f(x)=\tan^2 x$ and $g(x)=\sin^2 x$). However I am interested in how to find this answer. ...
1
vote
1answer
41 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
4
votes
2answers
97 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
3
votes
1answer
63 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
2
votes
1answer
79 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
17
votes
1answer
338 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
2
votes
2answers
46 views

why is it constant?

$f$ is a function from $\Bbb R$ to $\Bbb R$: $\frac{f(x+y)}{(x+y)} - (x+y)^2 = \frac{f(x-y)}{(x-y)} - (x-y)^2$ for all $x$ and $y$ the solution book just says "Thus $\frac{f(x)}{x} - x^2$ is a ...
1
vote
1answer
57 views

Proof read of functional equations

My teacher gave me this functional equation as an excercise $$f(x+f(y))=x+f(f(y))\,\, \forall\,\, x,y \in \mathbb{R}$$ If $f(2)=8$, calculate $f(2005)$ So my solution was For every $y$, let ...
0
votes
0answers
133 views

Solving an equation with 2 unknowns

I've been trying to solve this problem and was wondering if there is a more accurate / efficient way to do it. For the following equation $$y = a \times \left(1 - ...
6
votes
2answers
413 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
0
votes
0answers
100 views

integral equations - i need help to expand the function [duplicate]

I have the following integral equation to solve: $$\int_{0}^{2\pi} (\cos^2(x+y)+1/2) \phi (y) dy$$ So, I need to find $\lambda$ where $\lambda$ is the eigenvalues's function. Well, my main goal is ...
6
votes
2answers
226 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 $x=0$.
1
vote
2answers
61 views

$\frac{f(x_1)}{f(x_2)} = \log(\frac{x_1}{x_2}) \implies f(x)=\;?$

If $$\frac{f(x_1)}{f(x_2)} = \log\left(\frac{x_1}{x_2}\right),$$ what is $f(x)$? I mean the simplest form of $f(x)$, and what math technique you use to solve this problem? Thanks.
19
votes
3answers
948 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.
2
votes
1answer
91 views

How can I simplify this nasty equation between two functions?

I have the following equation: $$ h(n) = n \sum_{i=0}^{\lceil \log_2 n \rceil} \frac{m(2^i)}{2^i} $$ and I'm trying to understand exactly the relationship between the functions $h$ and $m$. The ...
5
votes
6answers
1k views

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
3
votes
2answers
346 views

How to find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$

Find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$. My solution: Let the first term of $P(x)$ be $ax^n$. We see first term of left side is easily $a^2x^{2n}$ ...
2
votes
3answers
336 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
5
votes
1answer
84 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) = ...
13
votes
4answers
777 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
6
votes
1answer
292 views

All functions $\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + \frac{2x^2}{f(y)}\right)$

How can I find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}^+$ such that $$\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + ...
1
vote
2answers
1k views

Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
15
votes
6answers
693 views

Solution to $1-f(x) = f(-x)$

Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$? I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...