# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### What are all polynomials $p(x)$ such that $p(q(x))=q(p(x))$ for every polynomial $q(x)$?

I assume that $p(x)$ and $q(x)$ are both real polynomials. If I let $q(x)=c$, (a constant) then $p(q(x)) = p(c) = q(p(x)) = c\ \forall c$. So $p(x)=x\ \forall x$. Is this operation valid and how ...
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### For what value of $c$ is $f$ periodic?

Let $f(x)=a\sin(cx)+b\cos(cx)$, where $a$, $b$ and $c$ are constants. Since $\sin$ and $\cos$ have a period of $2\pi$, if $c\in\mathbb{Z}$ then $f$ has a period of $2\pi$. How to prove the converse? ...
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### Would these witnesses satisfy this big-O function?

I'm trying to determine if $f(x) = \lceil x/2 \rceil$ is $O(x)$. I know that this is true, and the textbook answer is: $|\lceil x/2\rceil|\leq |(x/2)+1| \leq C|x|$ for all $x > 2$, with ...
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### Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
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### Domain and range of three functions [on hold]

Find the domain and range of: $f(x)=e^x-e$ $f(x)=\dfrac{\sqrt{x+1}}{2x}$ $f(x)=\dfrac{\ln(x)}{x} + 1$ Thanks.
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### Solve functional equation: f(x+y)=f(x)+f(y) [on hold]

Solve following functional equation: $$f:R\to R$$ $$f(x+y)=f(x)+f(y)$$
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### Is this true for functions with certain conditions?

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $f(x + y) + f(x − y) = 2f(x)g(y)$ for all $x$, $y$. Is it true that if $f(x)$ is not ...
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### Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
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### How do you evaluate this function? [on hold]

Given the function $g(x) = x^2 + 2x$, evaluate: $\displaystyle\frac{g(x)-g(a)}{x-a}$, where $x\ne a$ This is how far I got: $\displaystyle\frac{x^2 + 2x - a ^ 2 - 2a}{x-a}$, where $x\ne a$
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### Increasing function with $f'(x)=f(f(x))$ [duplicate]

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
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### Is this non-constant function periodic for every definable number?

Given the set $\mathbb{D}$ which contains all definable real numbers. The definition must not be infinite long. E.g. it contains $12$, $-3$, $\frac{1}{12}$, $\sqrt{2}$, $\pi^2$, $i+e$, Chaitin's ...
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### Functions invariant under scaling

Which functions are invariant under the transformation $$f(x)=af(bx)$$ for constants $a$ and $b$? Are functions of the form $cx^n$ and $de^x$ the only analytic ones (as in having a power series ...
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Well I was trying to find out whether the two sets $[n, n+1]$ and $[n, n+1]\cup \{n+2\}$ has the same cardinality. If you add another infinite set(not any random one) to $[n, n+1]$, for example $[n, n+... 5answers 61 views ### What is the domain and range of$y = \sqrt{9 − x^2}$? What is the domain and range of real function$f(x) = \sqrt{9 − x^2}$? In order to find the function's domain, you need to take into account the fact that, for real numbers, you can only take the ... 2answers 68 views ### Finding all possible values of a Function Let a function be defined as$f:N\to N$and$x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$and$1900<f(1990)<2000$. Find all values of$f(1990)$.$...
Is the hooked arrow map notation not supposed to mean an inclusion mapping? His definition is clearly showing inputs from $\mathbb{R}^2$ who's images are elements in $\mathbb{R}^{3}$ with fixed $z$ ...