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

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2
votes
2answers
44 views

How to determine $\mathbb{E}$ for which a maximum defined function is a bijection?

Assume that $f\colon\mathbb{E}\subset\mathbb{R}\rightarrow\mathbb{R}$, $f(x)=\max\lbrace2x-5,x-2\rbrace$. Determine $\mathbb{E}$ for which $f$ is a bijection. I was thinking it is ...
1
vote
3answers
191 views

A formula that yields a particular graph shape

I would like a formula for a function whose graph has the following properties: $f(0) = 0$. $\lim\limits_{x\to\infty}f(x) = y$. The shape of the function is approximately the following: It should ...
7
votes
2answers
475 views

Proving Injectivity

The problem is to show the function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by $$f(x,y)=(\tfrac{1}{2}x^2+y^2+2y,\,x^2-2x+y^3)$$ is injective on the set ...
0
votes
1answer
1k views

Mirror a function about x = c axis

I'm trying to mirror a function $f(x)$ about the $x=c$ axis. To mirror it about the $x=0$ axis you just have to plot $f(-x)$. I tried to mirror $f(x) = x^2$ about the $x = c$ axis. And I found that ...
1
vote
2answers
71 views

Reduction of $f:\mathbb{C}^n\to \mathbb{C}$ into sum of $f_{ij}:\mathbb{C}^2\to\mathbb{C}$

I was browsing wikipedia the other day when I came across the following (paraphrased) claim: $$ \exists f_{ij}:\mathbb{C}^2\to \mathbb{C} \mbox{ s.t. } f(x_1,\dots,x_n)=\sum_{i,j} f_{ij}(x_i,x_j) $$ ...
7
votes
1answer
145 views

$f(x)^2 ≥ f(x + y)(f(x) + y)$ for no $f$?

Prove that there is no function $f : \mathbb{R}^+ → \mathbb{R}^+$ such that $$f(x)^2 ≥ f(x + y)(f(x) + y)$$ for all $x, y > 0$. I can't think of a way of solving this.
1
vote
3answers
67 views

How to Calculate the function of an interval?

I need to calculate $f((1,4])$ for the function $$f(x)=x^2-4x+3.$$ The answers I can choose from are: a) [0,3] b) [-1,0) c) (0,3] d) [-1,3] e) (-1,0) f) (0,3) Can someone guide me? It may be ...
1
vote
1answer
68 views

Is there any word for a “regular scale”, as opposed to a “logarithmic scale”?

We all know what a "logarithmic scale" means. (It basically means that the distance between 1 and 10 is the same as the distance between 10 and 100, as shown on the figure.) However, what is the word ...
2
votes
1answer
176 views

An intuitive explanation for neural networks as function approximators ?

We use normal linear regression for modelling functions on datasets . But Can someone explain how neural networks help in approximating more complex ,especially non-linear functions ? intuitively , ...
0
votes
1answer
40 views

Finding the functions that describes a Viral effect

I can't seem to find a function that makes sense to me in this. Edit: Lets look at it like a party. Say there is a party in 4 weeks. 10 people are going to the party. They each invite 10 others to ...
3
votes
0answers
161 views

Does $\sqrt{\delta^2}$ make more sense than $\delta^2$?

What is the Product of $\delta$ functions with itself? was already asked some time ago. In a comment the OP states: I want to create $\delta(t)$ such that the product with itself is also ...
1
vote
1answer
186 views

Mapping from a rotational quaternion to a single angle

I have given an attitude quaternion that describes the rotation of a device. This roation is relative to the magnetic north, so the quaternion includes implicit information about the heading of the ...
2
votes
1answer
220 views

Prime Counting Function as Sum of Heaviside Step Functions

Inspired by Logarithmic derivative on the critical strip, I would like to ask, if it is possible to write $\pi(n)$ as a sum of step functions like the following: $$ \pi(n)=\sum_{k=1}^{N} H(n-p_k), ...
11
votes
5answers
1k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
3
votes
5answers
891 views

Why are zeros/roots (real) solutions to an equation of an n-degree polynomial?

I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that ...
2
votes
2answers
108 views

What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
votes
1answer
154 views

Function in $\mathcal{C}(X\times Y)$ can be “approximated” by sum of finite number of functions of the form $g(x)h(y)$

Let $X,Y$ be compact spaces if $f \in \mathcal C(X \times Y)$ and $\varepsilon > 0$ then $ \exists g_1,\dots , g_n \in \mathcal C(X) $ and $ \exists h_1, \dots , h_n \in\mathcal C(Y) $ such that ...
1
vote
3answers
336 views

set theoretic function, products of sets (product versus Cartesian product)

Regarding the products of functions in axiomatic set theory, two textbooks which I am reading (Halmos; Hrbacek/Jech) have said the following: "There is a natural one-to-one correspondence between ...
1
vote
0answers
74 views

Problem on the maximum of the function involving Stirling numbers of the second kind

When reading about the Coupon collector’s problem (CCP), I just came up with the following problem which I found very curious about, though it seems to have neither relation to the CCP nor practical ...
2
votes
1answer
61 views

Quotients and continuous maps

Suppose $f:\mathbb R\times \mathbb R\to \mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$ where the domain is given the usual topology and the latter the quotient topology. Why then is the restriction ...
0
votes
1answer
136 views

How can I know the two functions of the following lines?

I didn't use math since ages , now I am building a game (I am a programmer) and I need a mathematical function to draw the following 2 lines The red thick line goes from x=1 to x=10 (it may go ...
1
vote
1answer
3k views

Comparing the growth rates

How can I go about comparing the growth rate of the following functions? $$\sqrt n,\quad 10^n,\quad n^{1.5},\quad 2^{\sqrt{\log n}},\quad n^{5/3}.$$ I am looking for a more generic answer on how do ...
1
vote
0answers
243 views

Using Rouche's theorem

Let $p>1$. Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain. It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
2
votes
2answers
621 views

Parallel functions.

In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line. I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
2
votes
1answer
106 views

Problems on Schwartz Functions

(1) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (2) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (3) What ...
3
votes
1answer
797 views

Difference in limits because of greatest-integer function

A Problem : \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} results in the solution : 1 But the same function enclosed in a greatest integer function results in a 0 ...
0
votes
1answer
378 views

why function over just a relation?

Is there a real life example showing data that forms a relation is more useful than one that forms just a relation? what real life scenario motivates the extra condition on relation so we have ...
2
votes
2answers
534 views

If $ f(3x-1)= 12x+5$, what is $\ x \circ f(x)$?

Here, I am sharing just an example problem which is given in one of my textbooks: $$ \ \large{ f(3x-1)=12x+5 \ , \\ x \circ f(x)= \, ? \ } \ $$ And, on below of the question, the book has shown an ...
1
vote
1answer
62 views

Proof that infinite functions can fit a table of numerical values

Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal ...
1
vote
0answers
145 views

Math function for parabola

I need an implicit function that plots the parabola that I am showing you in the picture. Everything you need is shown there. The radius of the thickness of the parabola must be 3. Thank you in ...
2
votes
0answers
217 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
-1
votes
1answer
52 views

Domain of $f(x) < 3$ is $(0,\infty)$ and Domain of $f(x) > -1$ is $(-\infty , 5)$ [closed]

Domain of $f(x) < 3$ is $(0,\infty)$ and Domain of $f(x) > -1$ is $(-\infty , 5)$ What is the domain of $[f(x)]^2 \ge f(x) + 6$?
0
votes
1answer
79 views

Finding range of $f(g(h(x)))$

$$\begin{align*} f(x) &= \frac{2}{x+1}, \\ g(x) &= \cos x, \\ h(x) &= \sqrt{x+3} \end{align*} $$ Find the range of $f(g(h(x)))$. Please explain the problem.
5
votes
1answer
520 views

Who came up with the arrow notation $x \rightarrow y$?

I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it? Each map needs both an explicit domain and an explicit codomain (not just a domain, as in ...
1
vote
3answers
134 views

Mathmatical representation of recursion function

Well i'm not so good at math, but i have the following task: Here's the code: ...
1
vote
1answer
358 views

One sided limit of an increasing function defined on an open interval

Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$? If so, is it true that $\ell=\lim_{x\to ...
2
votes
1answer
169 views

Can the output of a mathematical function be another mathematical function?

Apologies if this is actually mathematical gibberish. I'm very familiar with mathematical functions at a simple level. A function relates a set of inputs to a set of outputs. What I'm trying to ...
0
votes
2answers
247 views

Find a function that fits data and has certain characteristics

I have some data. data = [ (10000, 0), (100000, 0.25), (10000000, 0.5), ] I want to find function(s) fitting this data. I have a possible starting ...
2
votes
0answers
113 views

Iterated Root Mean Square-Arithmetic Mean

Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible? if not possible, what is the closed form of it as ...
0
votes
1answer
172 views

How to combine ratings given by two different functions into one rating

I have a set of items: {s1, s2, s3}. There are two functions that rate these items: f1() and f2(). For example f1(s1) = 4. f1() returs values in [0,15], but f2() returns values in [0,100]. Moreover, ...
7
votes
2answers
312 views

Must $g$ be the identity if $f = g \circ f$?

I am finding it hard to solve the following problem. Let $A$ is a set and $f : A \rightarrow A$ and $g : A \rightarrow A$. If $f = g \circ f$, must $g$ be an identity function always? Will there be ...
2
votes
0answers
200 views

calculating the amplitude of a cosine function

I want to be able to be able to get the amplitude of the following function: $$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$ I am trying to find a way to get the ...
0
votes
1answer
522 views

Finding the Inverse of a Summation

I have seen more specific versions of this question but my question is more general. For any given summation does there exist an inverse. If not, how does one tell if the function has an inverse. ...
0
votes
1answer
228 views

function f satisfies f(xy) = f(x)/y , f(30) = 20. Find f(40)

The function $f$ satisfies $$f(xy) = \frac{f(x)}y$$ and $f(30) = 20$. Find $f(40)$.
2
votes
2answers
10k views

How to find the equation of a graph with given coordinates? [duplicate]

Possible Duplicate: Writing a function $f$ when $x$ and $f(x)$ are known If I am given 9 co-ordinates of a random graph say for e.g ...
3
votes
3answers
233 views

Inverse of $f^{-1}(x)=x^5+2x^3+3x+1$ question

Let $f$ be a one-to-one function whose inverse function is $f^{-1}(x)=x^5+2x^3+3x+1$. Compute the value of $x_0$ such that $f(x_0)=1$. I am confused as to what this question is asking me, ...
3
votes
3answers
122 views

What is an example of a linear function that maps a matrix to a scalar? What makes it a 'function'?

I suppose this is part terminology question and part math, but I am trying to untangle what we mean when we say "The linear function $f$ maps $R_{m\times n}$ space to $R_{m}$ space", and "The linear ...
0
votes
1answer
266 views

How to check if function is Lipschitz continuously differentiable?

I do have a problem with achieving convergence in Newton method (using Armijo rule) for system of algebraic non-linear equations. I suspect that my function is not continuously differentiable, however ...
1
vote
1answer
335 views

what is the difference between functor and function?

As it is, what is the difference between functor and function? As far as I know, they look really similar. And is functor used in set theory? I know that function is used in set theory. Thanks.
0
votes
1answer
102 views

Homotopy between $y=x$ and $y = {\rm step}(x-1)$?

I am looking for a homotopy between $y=x$ to $y = {\rm step}(x-1)$ (to the limit; a very steep sigmoid is also ok). More formally, I look for a family $$y_\epsilon \in C^1(A,A)$$ where $$A = ...