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

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0answers
18 views

'Function may not necessarily use all elements of specified range'?

In a textbook on the theory of computation, I encountered a passage which states: The outputs of a function come from a set called its range... In the case of the function abs, if we are working ...
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0answers
36 views

How to find the maximum of this function by observation: ${f}(x)=-124 \cos (x^{2})x^{2}-62 \sin (x^{2})$

How to find the maximum of this function by observation (without graphing)? ${f}(x)=-124 \cos (x^{2})x^{2}-62 \sin (x^{2})$ Using the fact that cos(x) and sin(x) is between -1 and 1.
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0answers
26 views

1. Determine which of the following functions are injective and which are sur- jective. [closed]

(a)$f:\mathbb{N}→\mathbb{N}, f(x)=3x+2$ (b)$f:\mathbb{Q}→\mathbb{Q}, f(x)=x3+2$ (c)$f:\mathbb{R}→\mathbb{R}, f(x)=−2x+5$ (d)$f:[−π,π]→[−1,1], f(x)=\sin x$. 22 (e)$f:\mathbb{R}→\mathbb{R}$, f(x)= x ...
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1answer
31 views

For $f(x, y) = x-y$, is $f(K \times K)$ closed if $K$ is closed?

$f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y) = x-y$. For $K \subset \mathbb{R}$ closed is $f(K\times K)$ closed? For the closed interval this is straight forwardly true ...
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2answers
68 views

Inverse of a function from $\mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ [closed]

Given the function $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ $f(a,b) = b^2 +a$, if $b>a$ or $a^2 +a + b$, if $b<a$, which associates $a,b \to z$, find its inverse, which ...
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1answer
20 views

Cardinality of the set of all indicator functions

Let's say we have two sets X={A,B,C} and Y={0,1}. We are trying to find the cardinality of the set of all functions from X to Y. From my understanding, this is supposed to be equal to the size of the ...
5
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1answer
129 views

Solving $f(x+y) = f(x)f(y)f(xy)$

Find all continuous functions such that $f(x+y) = f(x)f(y)f(xy)$ I solved it and got the two very obvious solutions, $f(x) = 0$ and $f(x) = 1$. Any other such functions?
0
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1answer
21 views

Looking for a function looking similar to quadratic function but skewed to left

I am looking for a function looking similar to quadratic function but skewed to left; something like this: Just ignore numbers and variables in the figure. I am interested only in the shape of the ...
1
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1answer
13 views

Find functions which change asymptotic properties if raised to 2

Kindly give an example of positive functions f(n) and g(n) such that f(n) = O(g(n)) but it does not hold that 2^f(n) = O(2^g(n)). A friend asked this question as this came in one of his ...
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1answer
39 views

Infinitely differentiable $f: \mathbb{R} \to \mathbb{R}$ such that for each $n \geq 0$, $f^{(n)}(x)=0$ if and only if $x=0$

I am looking for a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies these properties: i) $f$ is infinitely differentiable. ii) $f$ and all its derivatives should intersect the $x$-axis only at ...
1
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0answers
39 views

General definition of piecewise continuity

Is there a general definition of piecewise continuity for functions between topological spaces ? Of course one can intuitively says that $f: X \rightarrow Y$ is piecewise continuous if for every ...
0
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1answer
40 views

Matrices and bases

Can you please verify my argument: Let $M = \begin{pmatrix} a & b\\ c& d\end{pmatrix}$, where $a,b,c,d$ are all real. $$AM=\begin{pmatrix} c & d\\ a& b\end{pmatrix}$$ Let $B$ be ...
0
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1answer
16 views

²Inverse function of 2(x+1)/100 - x(x+1)/100² to build a linear distribution

What's the inverse function for y = 2(x+1)/c - x(x+1)/c² with c being a constant such a 100? I am building a linear probability ...
2
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2answers
116 views

Is the function $\,f(x, y) = x-y\,$ closed?

Is $f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$, such that $f(x,y)=x-y$ closed?
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1answer
14 views

Involutory function with positive derivative

I was looking at a situation where $v^{-1}(t) = v(t)$ and then $t = v(v(t))$. I started looking for solutions where the derivative $v'(t)$ is positive. The identity function $v(t) = t$ is a solution, ...
0
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1answer
19 views

Multivariable functions that are not continuous

Say we have a 2-variable function $f(x,y)$ which is not defined at $(x,y) = (0,0)$ (so perhaps a fraction of some sort). Say I wanted to make this a continuous function in $(0,0)$ by defining it to ...
0
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1answer
88 views

Problem in Basic Real Analysis

I want to prove that if $$E_{b,y} = \{ r\in \Bbb Q |\; b^r\leq y \}$$ for $b\in \Bbb R^{>1}$ and $y\in \Bbb R^{>0}$ then, $\sup E_{b,y}$ is the unique solution to the equation $$b^x = y$$ I ...
1
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3answers
19 views

Demonstrating that a function is monotonically increasing/decreasing

my question is more of a conceptual one, but i'll use the problem i'm stuck on to keep things clear. I am confused about how to demonstrate whether a function is strictly monotonically increasing or ...
0
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2answers
26 views

The cardinalities [duplicate]

I have some difficulties to resolve a problem. Could you explain me why this sets: $[a,b]=[c,d]$,where $a,b,c,d \in\mathbb{R}$ with $a<b$ and $c<d$ have the same cardinalities? thanks
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2answers
43 views

Find the functions $f$ that satisfy the given initial value problems

(a) $f'(x)+3x-2=0$, $f(2)=0$ (b) $2f'(x)-\sqrt{x^3} = 0$, $f(0) = 3$ I know the functions need to be integrated to find $f(x)$, however I am unsure as to how to integrate $f'(x)$ in the ...
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1answer
53 views

Showing a bijection without a specific function (and only with cardinality)

$\forall n \in \mathbb{N}$, define $I_n = \{k \in \mathbb{N} \mid k \leq n\}$. And for any set $A$, the number of elements in $A$ is $n$ if there exists a bijection from $I_n$ to $A$. Say the ...
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1answer
44 views

Determine the symmetry of $y=|x-4|$

Determine whether the graph of $y = |x − 4|$ is symmetric with respect to the origin, the $x$-axis, or the $y$-axis. A. not symmetric with respect to the $x$-axis, not symmetric with respect to the ...
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0answers
72 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
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1answer
32 views

Find a function where the mode is the minimum

Let $a_i\in\Bbb R$ some collection of data points where $0\le i\le n$. Define the function $$f(x)=\sum_{i=0}^n(x-a_i)^2$$ It is clear that the minimum value of $f$ occurs when $x$ is the mean of ...
0
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2answers
29 views

How does one compute the inverse of the function $f$ that satisfies $f(3x-2) = x-1$? [closed]

The problem is: Given $f: \mathbb{R} \to \mathbb{R}$ such that $f(3x-2) = x-1$, find $f^{-1}(x)$. It would be great if you could help me on this one
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1answer
48 views

How to show logistic function is monotonic increasing?

How to show logistic function is continuous monotonic increasing? $$\frac{1}{1+e^{-ax}}$$ Thanks in advanced..
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1answer
68 views

Minimization over two lines

This is a minimization question where the minimizing points can be chosen freely on two lines: $$\mbox{minimize}\, \prod_{i=1}^K {y_i}\quad \mbox{such that}\quad \prod_{i=1}^K ...
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0answers
39 views

On the limit of a special kind of function

Let $(a_n)$ be a strictly increasing sequence such that $ \sum \dfrac 1{a_n}$ converges ; then off-course $\lim \bigg(\dfrac 1{a_n}\bigg)=0$ , so $\lim (a_n)=\infty$ , in particular $(a_n)$ is ...
2
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3answers
33 views

If $\lim\limits_{x \to 1}g(x) = 0$ and $\lim\limits_{x \to 1}f(x)=1 $, Then$\lim\limits_{x \to 1}\dfrac{f(x)}{g(x)} $ does not exist.

If $\lim\limits_{x \to 1}g(x) = 0$ and $\lim\limits_{x \to 1}f(x)=1 $, Then$\lim\limits_{x \to 1}\dfrac{f(x)}{g(x)} $ does not exist. iS this statement true. I believe this is not true. The main ...
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3answers
135 views

determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.

I am new to multivariable calculus and my textbook doesn't give out solutions so I'm just wondering how you go about proving something like this? I know that a function is differential at a point $a$ ...
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1answer
33 views

Calculate winner of soccer match

I am writing a program that simulates a soccer tournament between countries using their FIFA rankings. I am looking for a function that takes two country rankings and outputs a number between (about) ...
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1answer
36 views

Is there an infinitely differentiable function such that $f$ intersects the $x$ axis only at the origin and no derivative of $f$ is $0$?

Is there an infinitely differentiable function $f$ such that $f(x)=0 \iff x=0$, $f$ is infinitely differentiable No derivative of $f$ is ever equal to zero.
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2answers
51 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
0
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0answers
24 views

Function representation and meaning…

I have such sort of a question: I do not understand the following function, or representation of the function $f$: $$f:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$$ How can it be that the input ...
0
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2answers
21 views

Discriminating Function

Does any function $f(x)$ exists? such that $$ f(x) = \begin{cases} a , & \text{if $x$ is rational} \\ b , & \text{if $x$ is irrational} \end{cases} $$ where $a \neq b $
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1answer
58 views

complex differentiation

It is my first text here. So I have started to look at complex numbers in death. I do Uni know, so adding $3+4i$ and $4+7i$ is now nothing. What I am stuck on is the idea of taking a derivative of a ...
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2answers
59 views

How to find sum of coefficients of $\frac{d^n}{dx^n} \left( Z(x)^m \right)$

I noticed something interesting result but I do not know how to prove it (or disprove) Function $U$ defined as $$ U(Z(x),Z'(x),Z''(x),Z'''(x),...,Z^{(n)}(x))=\frac{d^n}{dx^n} \left( Z^m(x) ...
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0answers
19 views

Prove that the composition of two “closed form functions” is itself a “closed form function”?

So I have been given the definition of a "closed form function" that is a set of functions built inductively (mapping from and to the complex) starting with the fact that the constant functions ...
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1answer
472 views

Function that returns negative number for negative x and y

I need a function, $$f(x,y):\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $$, that returns negative a number only if both x and y are negative. It can use only the four basic operations, so for ...
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1answer
10 views

a positive interger solution for a fraction of logarithm

Let $r_1,r_2,n$ be positive integers with $r_1<n,r_2<n$ and $r_1\neq r_2$. Given $$ n=\frac{r_1\ln(r_1+1)-r_2\ln(r_2+1)}{\ln(r_1+1)-\ln(r_2+1)}. $$ My question is: Can you find $r_1,r_2$ and $n$ ...
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0answers
21 views

Logarithm Function

Can you draw a graph for this function Df: x>3/2 Points: 3-2x=1 x=1, y=-1, P1(1,-1) 3-2x=1/2 x=5/4, y=-2, P2(5/4,-2) x=0, y=0,585, P3(0,0.585)
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1answer
15 views

Is there exists other statements equivalent to the analytic rank?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The order of vanishing (the analytic rank) at a point $s=a$ is denoted by $m$ (the ...
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1answer
64 views

Example of an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 and f intersects origin with infinite multiplicity

Is there an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 for which it is reasonable to say the graph of f intersects the x-axis at the origin with infinite multiplicity. So ...
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2answers
32 views

Horizontal Asymptote of Strange Function

What is the horizontal asymptote as x approaches positive infinity of $\sqrt{4x^2 + 5x} - \sqrt{4x^2 + x}$? The horizontal asymptote is in the form $y = k$. Find $k$.
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1answer
12 views

Can anyone explain how to show the finite difference equation $y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$?

I was given that $y_{j}=y(x_{j})$ where $x_{j}=x_{0}+jh$ for integer j and positive h. I need to show that $$y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$$ I thought I could start by finding the Taylor ...
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2answers
21 views

Let $f:R \to R$ be a function with $\frac{f(x)f(y)-f(xy)}{3} = x+y+2$ for all real numbers $x,y$. List all possible values for $f(36)$.

So far I have just been plugging in possible $x$ and $y$. $$\frac{f(4)f(9)-f(4\cdot9)}{3}=4+9+2$$ So then $f(36)=f(4)f(9)-45$. $$\frac{f(6)f(6)-f(6\cdot6)}{3}=6+6+2$$ ...
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2answers
16 views

function with moving average on matlab

Do a function that maintains moving average, the function gives the average of all the numbers that have been put in the function. does anyone know how to do this ?
2
votes
4answers
83 views

value of $x^2 \sin (\frac1x)$ at $x=0$ [closed]

What is the value of $y=x^2 \sin(1/x)$ at $x=0$ $x^2 =0$ but $\sin (1/x)$ is undefined in general if $y= a(x)b(x)c(x)d(x)\dots$ i.e a function made up of a product of functions At a specific ...
0
votes
1answer
36 views

Functions involving codomains

Problem: Consider the possible $f: [7]\to[9]$ a) How many have $f(i) $even , for all i? b) How many have rng(f) = {5,6} As far problem a goes, I've only gotten to the answer = 4^7. However I'm not ...
0
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2answers
19 views

Help with demonstration of formula for the axis of a parabola

At school we are studying the parabola and our teacher said that the formula for the axis of a parabola is $x=-\frac{b}{2a}$ without giving us the demonstration; so I tried to come up with a nice ...