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

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0
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1answer
48 views

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 ...
0
votes
0answers
34 views

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? ...
-4
votes
1answer
40 views

Piecewise Contextuous Functions - A possible new branch of functions?

Let us define f(x) as the following: $$f(x) = \begin{cases} g(x) & \text{if f(x) is being floored ($\lfloor x \rfloor$)} \\ h(x) & \text{if multiplication by anything other than 1 is being ...
2
votes
2answers
2k views

Trying to figure out a formula with given input and outputs.

I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers. ...
0
votes
1answer
22 views

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'?
0
votes
4answers
67 views

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 ...
-1
votes
1answer
38 views

Are there mathematical objects (like matrices) which behave like shorthand operators for complicated calculations? [on hold]

Matrix multiplication involves summing a product. It is appropriate where you need to multiply things together and then add. Are there more examples like this ? Namely, to use a mathematical object, ...
0
votes
1answer
455 views

How to solve for maximum area of a rectangle under a curve?

Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is $8^{-\frac{x}{5}}$. I would greatly appreciate a full explanation. I already ...
1
vote
2answers
39 views

$A : \mathbb{R}^n \to \mathbb{R}^n \implies A(\mathbb{Z}^n)=\Gamma$ on theTorus

In the Analysis on Manifolds via the Laplacian page $51$, it is indicated that if $A : \mathbb{R}^n \to \mathbb{R}^n$ be so that $A(\mathbb{Z}^n)=\Gamma$, then $\text{Vol} (\mathbb{T})=\det A$. Could ...
4
votes
0answers
27 views

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 $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$
-1
votes
0answers
21 views

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 ...
3
votes
2answers
110 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as summation,...
4
votes
2answers
41 views

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 ...
0
votes
2answers
24 views

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.
1
vote
1answer
43 views

If y is not an exterior point of $K$, then there exists a $x$ in $K$. Is it true?

For a vector $v = (x_1,\ldots,x_d)^t \in \mathbb{R}^d$, we let the function $f$ be $f(v)=|v|^2=v^tv=x_1^2+\cdots+x_d^2$. Is it possible to show that there exists a x $\in K$ which satisfies $f(x)>...
2
votes
1answer
36 views

Interpolation for $f(n),n\in\mathbb{Z}$: Does it converge?

Assume a function $f(n)$ which is defined for $n\in\mathbb{Z}$. For each period $[n,n+1]$ the function could be interpolated with a polynomial of degree $m$. The polynomials should be built in a way ...
1
vote
3answers
39 views

Is the function $\frac { x-1 }{ \ln { { \left( x \right) }^{ 2 } } } $ continuous at$ x=0$?

I would like to know whether the function shown in the title is continuous or not at $x=0$. This problem is disturbing since the function isn't defined at x=0, but the limit of the function as x ...
1
vote
2answers
32 views

Minimizing basic boolean function

$==============================$ Given the function $f(x,y,z) = y'z'+x'y+x'yz+xyz'$ (where ' means the NOT operator), I need to transfer this function to its basics. The possible answers are: $x'y+y'...
0
votes
0answers
12 views

Polynom subspace of continuously differentiable Functions

Let $n\in \mathbb{N}$ and $a\in \mathbb{R}$. Then $\mathcal{C}^n(\mathbb{R})=:V$ and $$\langle f,g\rangle :=\sum_{k=0}^n {f^{(k)}(a)g^{(k)}(a)}$$ is a positive semidefinite Bilinear Form for all $f,g\...
0
votes
3answers
40 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
3
votes
4answers
73 views

Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$ For ...
1
vote
0answers
8 views

Necessary and sufficient condition for argmax/argmin

Let $x_1,\dots,x_n$ be real variables and let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a differentiable function with a unique maximum (or mininum). Is there a necessary and sufficient condition ...
0
votes
1answer
11 views

Etymologies of injections and surjecteions

Why one-to-one functions are called "injections" and onto functions are called "surjections"?
-1
votes
3answers
59 views

Is function $F(x)= 2x^2 -3x$ increasing or decreasing [on hold]

$F(x)= 2x^2 -3x$. find the range of $x$ to check whether the function the is strictly increasing and strictly decreasing.
0
votes
2answers
25 views

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 ...
3
votes
1answer
10 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
2
votes
2answers
51 views

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
1
vote
2answers
34 views

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 ...
-1
votes
1answer
79 views

How to find the function such that $\int_0^1f(x)\ \mathrm dx=e^{-4n^{2}{\pi}}$ [on hold]

Find $f(x)$ where: $$ \int_{0}^{1}f(x,n)\ \mathrm dx=e^{-4n^{2}{\pi}} $$ Is it possible that question contains infinitely many answers? How to solve this ? Please provide me a hint.
1
vote
4answers
217 views

why $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?

Why does $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$? Breaking up the limit: $\lim\limits_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3 $\lim\limits_{x\to-\...
1
vote
3answers
24 views

Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$

Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$ Here's what I did: $$A = \lim_{n \to +\infty} \sum^{n}_{i=1}f(x_{i-1})\Delta x \\ = \lim_{n \to +\infty} \sum^{n}_{i=1} 2(i-1)...
0
votes
1answer
36 views

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 ...
3
votes
4answers
87 views

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 ...
-4
votes
1answer
15 views

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.
-5
votes
0answers
46 views
7
votes
1answer
34 views

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 ...
0
votes
2answers
50 views

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 ...
1
vote
1answer
12 views

Inverse Function of a Closed Interval

This is an example from the book Topology 2nd edition by Munkers. Example 4 page 20. Please see the example here How is it that $f^{-1}(2) = -1$? Shouldn't $f^{-1}(2) = 0$? Since $f(x) = 3x^2 + 2 \...
-7
votes
1answer
55 views

Help Me Solve A Few Problems [on hold]

Here is it $$2y^2y'-x^2=0,\quad y=\text{?}$$ For example $$y^2y'-5x=0$$ $$y^2 \,dy -5x \,dx = 0$$ $$y^2\,dy=5x\, dx$$ $$\int y^2\, dy = \int 5x\,dx$$
0
votes
1answer
9 views

How to translate a function along its normal in any given point?

Let f be some differentiable function in R. Let n(x) be the normal of the tangent at (x, f(x)). How can you translate each point of f along the normal at this point given a certain distance?
1
vote
0answers
38 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
0
votes
1answer
45 views

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 $x\ne a$
5
votes
2answers
81 views

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 $x$?
1
vote
1answer
35 views

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 ...
1
vote
1answer
43 views

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 ...
1
vote
2answers
20 views

Do both sets have the same cardinality?

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+...
0
votes
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 ...
1
vote
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 $f(1990)$. $...
0
votes
0answers
31 views

Inclusion Mapping?

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 $z$ ...
1
vote
1answer
26 views

How do I “stretch” and “compress” a piecewise function?

I have Googled a few times and experimented on Desmos, but both attempts were to no avail, and now I come here. How is a piecewise function transformed to be "stretched" or "compressed"? What about ...