# Tagged Questions

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### At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
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### 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, ...
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### 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 $m$ ...
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### Find $f(x)$ where $f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $f(x)+f\left(\frac{1-x}x\right)=x$ ?
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### 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 ...
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### 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 ...
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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 $$y = a \times \left(1 - ... 2answers 421 views ### Find all f(x) if f(1-x)=f(x)+1-2x? To find one solution I assumed that f is even and rewrote this as f(x-1)-f(x)+2x=1. By just thinking about a solution, I was able to conclude that f(x)=x^2 is a solution. However, I am sure that ... 2answers 230 views ### Find all function f:\mathbb{R}\mapsto\mathbb{R} such that f(x^2+y^2)=f(x+y)f(x-y). Find all function f:\mathbb{R}\mapsto\mathbb{R} such that f(x^2+y^2)=f(x+y)f(x-y). Some solutions I found are f\equiv0,f\equiv1, f(x)=0 if x\neq0 and f(x)=1 if x=0. 2answers 61 views ### \frac{f(x_1)}{f(x_2)} = \log(\frac{x_1}{x_2}) \implies f(x)=\;? If$$\frac{f(x_1)}{f(x_2)} = \log\left(\frac{x_1}{x_2}\right),$$what is f(x)? I mean the simplest form of f(x), and what math technique you use to solve this problem? Thanks. 3answers 1k views ### Given f(f(x)) can we find f(x)? Given f(f(x))=x+2 does it necessarily follow that f(x)=x+1? This question comes from a precalculus algebra student. 1answer 91 views ### How can I simplify this nasty equation between two functions? I have the following equation:$$ h(n) = n \sum_{i=0}^{\lceil \log_2 n \rceil} \frac{m(2^i)}{2^i} $$and I'm trying to understand exactly the relationship between the functions h and m. The ... 6answers 2k views ### What function satisfies x^2 f(x) + f(1-x) = 2x-x^4? What function satisfies x^2 f(x) + f(1-x) = 2x-x^4? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution. 2answers 347 views ### How to find all polynomials P(x) with real coefficents satisfying P^2(x)-1=4P(x^2-4x+1) Find all polynomials P(x) with real coefficents satisfying P^2(x)-1=4P(x^2-4x+1). My solution: Let the first term of P(x) be ax^n. We see first term of left side is easily a^2x^{2n} ... 3answers 338 views ### Finding an f(x) that satisfies f(f(x)) = 4 - 3x I need to find f(f(x)) = 4 - 3x In other examples, such as f(2), I can see that the result equates to -2 or f(x^2) becomes -3x^2 + 4. Do I really just substitute f(x) for x and ... 1answer 88 views ### Finding a real value of p I am a bit confused about approaching this problem, Let g(x) be a function such that g(x + 1) + g(x − 1) = g(x) for every real x. Then for what value of p is the relation g(x + p) = ... 4answers 790 views ### Solving for the implicit function f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right) and f(1)=1 How can I find all functions f:\mathbb{R}\to\mathbb{R} such that f(1)=1 and$$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$for all real numbers x and y with y\neq0? PS. This is ... 1answer 293 views ### All functions \frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + \frac{2x^2}{f(y)}\right) How can I find all continuous functions f:\mathbb{R} \rightarrow \mathbb{R}^+ such that$$\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + ...
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 ...
### 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 ...