Tagged Questions

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

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Determining domain and range

Intro I am trying to find a systematic method of finding the domain and range of a function. If I do find a successful method, I could potentially make a computer program that calculates domain and ...
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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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How to find vertice by two angles and side?

I know 'alpha', 'betta', length 'c', vertices: 'A' and 'B' How can i find the 'C'?
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How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} I managed to solve one part of this problem but that one part is ...
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sufficient reason for a function to be bijective

I know of course that an application $\Phi: A \rightarrow B$ is bijective if it is injective and if it surjective. I also know that for all bijective function, there exists an inverse. My question ...
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What is the name for a function that behaves symmetrically when its arguments are scaled?

In other words, is there a name for this property of a function $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$
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If $f: A \rightarrow B$ is surjective, and $A, B$ are nonempty sets, and $X \subseteq A$, does $f(A) - f(X) = f(A - X)$?

I'm working on a proof, and the proof will be complete if this is true... but I can't find a theorem in my book that explains whether or not this is true.
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How can I prove or disprove that there exists a function such that…

Suppose we have a function $f$ of $bx-ay$ where $a$ and $b$ are two real constants, if we have for example $e^{bx-ay}$ then obviously it is a function of $bx-ay$. Can we find a function $f$ such ...
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What is the meaning of this theorem regarding periodic functions?

I recently got acquainted with a theorem: If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$. I am having a difficulty in ...
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How to calculate inverse of $y=3x+4\log(x+1)$?

How to calculate inverse of $y=3x+4 \log(x+1)$? Wolframalpha says that http://m.wolframalpha.com/input/?i=Inverse+3x%2B4+log%28x%2B1%29+&x=0&y=0
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Existence of Continuous bijective function

I was looking for examples of bijective continuous functions infollowing cases Does there exist Bijective continuous function $(0,1) \rightarrow \mathbb{R}$? Yes, $f(x)=\frac{2x-1}{x-x^2}$ Does ...
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Composite functions yeah

I'm trying for the GRE so that I can apply for grad school in 2017. I've been working well through calculus and algebra. I'm making good progress but functions has been a challenge. Take this for ...
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Function output by parameter - value relations

I have a parameter set, P; p1, p2 ... pn And also I ...
The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
I have $f(x)$. I know this $$|f'(x)|≤ C < 1$$ $$0≤C$$ $$x ∈R$$ and I have this sequence $$a_1=x\\ a_2=f(x)\\ a_3=f(f(x)),\\\vdots\\ a_n=f(\cdots(f(x))\cdots)$$ I need to prove this sequence ...