# Tagged Questions

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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### A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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### When is it possible to have $f(x+y)=f(x)+f(y)+g(xy)$?

According to the question mentioned here, it seems that there is no function $f(x)$ such that the functional equation $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ can hold. Motivated by this question, I found it ...
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### New SAT Math Section: Comparing Equation of Line to Graph

This is a math question on a practice test for the New SAT that will come out in March. These questions should not go above the level of precalc. I'm posting a picture of the problem as well because ...
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### Functional Equation: When $f(x+y)=f(x)+f(y)-(xy-1)^2$

How does one solve the following functional equation when $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ When I assumed it was a polynomial equation, it can be seen through ...
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### General solution of recurrence relation [closed]

I am supposed to solve for the general solution of $f(n+2)=2(f(n+2))^2 -f(n+2)f(n)-2012$. I tried the method of generating functions but I am stuck with the power $2$ on the RHS. any other methods or ...
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### (Non-continuous) solutions to $f(f(x))=kx$ and $f(x^2)=xf(x)$

Given a fixed non-zero constant $k\in\mathbb{R}$, find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $$f(f(x))=kx\quad\text{and}\quad f\left(x^2\right)=xf(x).$$ If $f$ is continuous, then ...
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### If $f(x-2)=x$ for all real numbers x, then what is $f(x)$?

If $f(x-2)=x$ for all real numbers x, then $f(x)=?$ I think the answer stays the same, because the given says for all real x. so is $f(x)=x$ or i am wrong?
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### If $f(xy) = f(x) + f(y)$, show that $f(.)$ can only be a logarithmic function. [duplicate]

As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.
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### Show that $f(x^a) = a f(x)$ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1}$ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
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### if $\ f(f(x))= x^2 + 1$ , then $\ f(6)=$?

I want to know how to solve this type of questions. How can I find $\ f(x)$ from $\ f(f(x))$ Suppose, $\ f(f(x)) = x$ , then $\ f(x)=x$ or $\ f(x)=\dfrac{(x+1)}{(x-1)}$ how to find these ...
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### Find all the function that satisfy $f(x+y)+1=f(x)+f(y)$

Let function $f:R\setminus 0\to R$ such (1): $$\dfrac{f(x)}{x}=f\left(\dfrac{1}{x}\right),\forall x\neq 0$$ (2): for any $x,y$ such $$f(x)+f(y)=f(x+y)+1,\forall x+y\neq 0$$ Find $f$ Let $P(x,y)$ be ...
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### Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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### find all functions satisfying the condition. [duplicate]

Find all functions $f:\mathbb{R}\to{\mathbb{R}}$ such that $f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)$ for all $x,y\in{\mathbb{R}}$ first I put $x=y=0$ so I got $f(0)=0$ or $f(0)=2$. For the case $f(0)=2$, ...
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### Showing an equation with integrals of sinus [duplicate]

I have to show the following equation: $\int_{0}^{\pi} t \cdot f(sin \; t) \; dt = \frac{\pi}{2} \int_{0}^{\pi} f(sin \; t) \; dt$ with $f : [0, 1] \rightarrow \mathbb{R}$ is continuous. I ...
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### Not Understanding a specific substitution rule

I was given the question, If $f(3x+5) = x^2-1$, what is $f(2)$? I am trying to understand the reasoning why $3x+5$ is set equal to $2$.
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### Solving functional equation $f(x+y)=f(x)+f(y)+xy$

We are given $f(0)=0$. Then when $x+y=0$: $$0=f(-y)+f(-x)+xy$$ Can I now use $x=0$ and obtain: $$0=f(-y)?$$ Is this correct? Is there a better way to solve this equation?
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### Functional equation $f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$ with $f'(1)=1/2$

Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.
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### Solve $g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$

Let $y$ be a real number. Find $g$ such that $$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$ Is valid for all real $x$.
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### Let $f:\mathbb{R}^m \to \mathbb{R}$ be differentiable s.t. $f(x/2)=f(x)/2, \forall x \in \mathbb{R}^m$. Show that $f$ is linear.

Basically, I am not really sure how to start. I thought about going through induction to show for $\mathbb{N}$ and $\mathbb{Q}$, then use the completeness of $\mathbb{R}$, but I think it is a long ...
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### About $f(x)= f(\frac{1}{x})$

Consider the equation $$f(x)=f\left(\frac{1}{x}\right)$$ Where we want $f$ to be real-meromorphic. Are all solutions $f$ of the form $$f(x) = g\left(\frac{x}{1+x^2}\right)$$ Where $g$ is a ...
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### How can one show that $f(0)=0$ for $f$ satisfying certain conditions?

Given the functional equation $$f(x+(1+x)f(y))=y+(1+y)f(x)$$ Such that $f:(-1,\infty) \to (-1,\infty)$ and the function $g(x):=\frac{f(x)}{x}$ is strictly increasing in $I=(-1,0)\cup(0,+\infty),$ ...
### $3^x-2^y=1$, $x \in \mathbb{N}$ and $y \in \mathbb{N}$
$3^x-2^y=1$ or $y=\log_2{\left(3^x-1\right)}$ $x$ and $y$ must be natural numbers. I know this two solutions: $x=1$ and $y=1$ $x=2$ and $y=3$ Are there more solutions? How can I find them?
### Verify proof of $f(x)=e^x$ if $f(x+y)=f(x)f(y)$ and $f'(x)$ exists for all $x$
This is exercise 6.26.8 from Tom Apostol's Calculus I, I'd like to ask someone to verify my proof. I'd be also interested in alternative proofs: If $f(x+y)=f(x)f(y)$ for all $x$ and $y$ and if \$...