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

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0
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2answers
24 views

Can $y$ be made the subject in such equation?

I came across the equation: $${x^2} + 2xy + {3y^2} = 1$$ I tried making $\space y \space$ the subject, but the furthest I got was: $$ 2xy + {3y^2} = 1 - {x^2} $$ $$ y(2x + 3y) = 1 - {x^2} $$ $$ y ...
0
votes
0answers
29 views

Lebesgue measurable function equals Borel measurable function a.e.

I am trying to prove that if $f: \mathbb R^n \to \overline{\mathbb R}$ is measurable, then there exists $g: \mathbb R^n \to \overline{\mathbb R}$ Borel measurable such that $f=g$ a.e. I know this ...
1
vote
1answer
51 views

Functional equation - nonlinear

How do I go about solving this functional equation? $ 2f(n) = 2n^2 + f(2n) $, where f(n) takes integervalues.
3
votes
1answer
48 views

Given bijection between $\mathbb{N}$ and $A$ and $B$, find bijection from $\mathbb{N}$ to $A \cup B$

Let $A$ and $B$ be two countable sets and consider that $f$ is a bijection from $\mathbb{N}$ to $A$ and $g$ is a bijection from $\mathbb{N}$ to $B$. I have to find a bijection from $\mathbb{N}$ to $A ...
0
votes
3answers
46 views

Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
1
vote
1answer
75 views

Problem involving polynomial function and prime numbers

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
1
vote
1answer
43 views

Difference between two zero-one indicator functions

I have a zero-one indicator function $I(cond)$ which returns $1$ if the condition cond is true and $0$ if cond is false. Now I have the following difference: $I(a = c) - I(b = c)$. For some reason, I ...
0
votes
1answer
102 views

Find a positive convex function $f$ defined on $[a,b]$, s.t. $f(a)\times f(b)=1$ and $\int_a^b{f'^2dt}=12$

Find a function $f:[a,b]\to \mathbb{R}$ which is convex on $[a,b]$ such that $\int_a^b{f(t)dt}=0$, $\int_a^b{f'^2(t)dt}=\frac{12}{b-a}$, and $f(a)f(b)=1$? Another similar question which states: Find ...
0
votes
1answer
26 views

Finding a function with given partial derivatives dx dy

I need to find a function $f(x,y)$ such that $f(x,y)dx = \frac{1}{2}\frac{x}{\sqrt{x+y}}$ and $f(x,y)dy = \frac{1}{2}\frac{y}{\sqrt{x+y}}$ how can this be solved?
3
votes
2answers
83 views

What is implied by $f \circ g = g \circ f$?

For any two functions $f(x)$ and $g(x)$ we are given $f \circ g = g \circ f$. What does this imply? I found that $f(x) = g(x)$, $f(x) = g^{-1}(x)$ and $ f(x) = x \ (\neq g(x))$ are some of the ...
0
votes
0answers
23 views

Topology: every continuous function on $\mathbb{R}^2$ scales a point

The question is simple: Suppose $f : \mathbb{R}^ 2 \to \mathbb{R}^ 2$ is continuous. Show that there exist $\lambda > 0$ and $x \in \mathbb{R}^2$ such that $f(x) = \lambda x$. So basically, we ...
3
votes
2answers
33 views

is $y = \sqrt{x^2 + 1}− x$ a injective (one-to-one) function?

I have a function $y = \sqrt{x^2 + 1}− x$ and I need to prove if it's a Injective function (one-to-one). The function f is injective if and only if for all a and b in A, if f(a) = f(b), then a = b ...
0
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0answers
13 views

generate unpredictable value with a shared key

I will start with an example. Some beacon like estimote Beacons use something called secure ID. The ID transmitted by the beacon change every 10 minutes autonomously without contacting the server. ...
-3
votes
0answers
29 views

For which $c>0$ is $c^{\ln x}$ strictly decreasing, increasing and limited? [closed]

Let $f(x)=c^{\ln x} : (0,\infty) \to \Bbb R$. For which $c>0$ is $f$ strictly decreasing, increasing and limited? How can I examine that? Please don't tell me to use Wolfram Alpha or a ...
1
vote
1answer
36 views

What is the term for relation whose inversion is a function?

Do we have a conventional term/name for such a relation $R$ (which is not necessarily a function) that $R^{-1}$ is a function? If not, what are your suggestions?
0
votes
2answers
25 views

linear transformation and surjective function

Good night, i have an big dude with this How i can know if an linear transformation is surjective? because for example i know if an function is injective if kernel is 0 but with the surjective ...
2
votes
5answers
103 views

Find the range of the function $y = \sqrt{x^2 + 1}− x$?

I have a function $y = \sqrt{x^2 + 1} − x$, where the Domain is $(−\infty,+\infty)$. Explanation for the domain I need to make sure the domain of the function does not include values of $x$ that ...
9
votes
2answers
124 views

Is a function of $\mathbb N$ known producing only prime numbers?

It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$. What about the function $f(n)=a^n+b$ ? Do positive integers $a$ and $b$ exists such ...
1
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1answer
41 views

Formal Notation for Finding Inverses of Functions

Generally in most introductory university courses, finding the inverses of functions, is done in what seems to be to be a very haphazard way. Given any scalar function $f : \mathbb{R^n} \to ...
0
votes
1answer
19 views

Prove that $\exists k \in \mathbb{N}$ such that $k\leq m$. And such that exists Bijective function $g:A \rightarrow \mathbb{N}^{<k}$.

$k \in \mathbb{N}$ $\mathbb{N}^{< k} = \{ j\in \mathbb{N} |j<k \}$ Let A be a set,Assume $m \in \mathbb{N}$ such that exists Injective function $f:A\rightarrow \mathbb{N}^{<m}$ ...
0
votes
3answers
20 views

What is the Order of operations for finding the inverse of a function AND solving.

I have $y=4(x+2)^3$. So first part of taking the inverse is switching the variables $x$ and $y$ so you'd have $x=4(y+2)^3$. Why does the exponent $3$ get put in front of the square root symbol? The ...
0
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0answers
36 views

Holomorphic function $f(\frac{1}{2n})=f(\frac{1}{2n-1})=\frac{1}{n}$

Does there exists a holomorphic function $f:B_2(0)\to\mathbb C$ such that $$f(\frac{1}{2n})=f(\frac{1}{2n-1})=\frac{1}{n},\forall n\in\mathbb Z^+$$ I do not have an idea?
12
votes
3answers
314 views

$f(nx)\to 0$ as $n\to+\infty$

Let $f:\mathbb R^+\to\mathbb R^+$ be a continuous function, and let $I$ be a subset of $\mathbb R^+$ such that the following property holds: For any $x\in I$, $f(nx)\to 0$ as $n\to+\infty$. ...
0
votes
1answer
25 views

How to find a function on a 3-dimentional space base on a set of points?

So I have been working on stereographic projections and stuff. I want to input a set of points on the plane z = -1 and find the function on the sphere that it was projecting from. Right now I have ...
-1
votes
1answer
16 views

Relationss and functions

I have been given a question based on math I don't know much about. is it possible someone could give me the answer and run me through it so I can have an understand on how to do this, it would be ...
1
vote
0answers
21 views

Will this function be odd?

Question: If $f:R\to R$ is an invertible function such that $f(x)$ and $f^{-1}(x)$ are symmetric about the line $y = -x$, then: A) $f(x)$ is odd B) $f(x)$ and $f^{-1}(x)$ may not be ...
2
votes
1answer
53 views

What do you call a function $f$ such that $f(f(x))=x$?

What is the name of the property of a function that yields the original result when done twice in a row: $$f(f(x)) = x?$$ I'm pretty sure there is a word for these functions, but I haven't been able ...
0
votes
0answers
22 views

Monotonic and differentiable function

Question: $f: R\to R$ is a differentiable and monotonic function such that $f(f(x)) = k(x^{11} + x), (k \neq 0)$. Find the values that $k$ can take. Differentiating the given expression: ...
1
vote
1answer
24 views

Double dual of $\mathcal{l}^1$

A little while ago, I asked this question and it all got sorted out. But, now I am asked to show that $J$ is not onto, if $\Omega = \mathcal{l}^1$. We haven't learned anything about reflexive spaces, ...
17
votes
3answers
2k views

How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
2
votes
0answers
30 views

Function that maps 'maximum metric-based' distance function to 'euclidean metric-based' distance function

Let's define a 2-vector: $$ v \in \mathbb{R}^2,\ \ \ v=[v_x,v_y] $$ We then have a 'maximum metric-based' 'distance + rotation' function $d_{max}(v)=[|v_x|+|v_y|,v_{\theta}]$ and a 'distance + ...
6
votes
2answers
86 views

Prove $1+a<b+c$ given $a>0,b>0,c>0$, $a<bc$ , $1+ a^{3} = b^{3} + c^{3} $.

Suppose: $a>0,b>0,c>0$, $a<bc$ , $1+ a^{3} = b^{3} + c^{3} $. Prove: $$ 1+a<b+c $$ This inequality at the University of Toronto plan.
1
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0answers
34 views

Notation in Double Dual Mapping

I am quite confused at the notation used for $J$ in the following question: For a normed space $\Omega$, let $\Omega^{**}$ denote the dual space of the dual space. Let $J: \Omega \rightarrow ...
0
votes
1answer
35 views

Is there an opposite of the Kronecker Delta?

Instead of $\delta(n,n) = 1$ and $\delta (n,k) = 0$, is there something that returns $0$ when the arguments are the same, and $1$ when the arguments are different. Is there a special function that ...
0
votes
2answers
45 views

A function that maps $R^m \to R^n$ where, $m<n$ is called?

I forgot the name of the type of functions that map a smaller space to a larger one, in this case $$g(x_1,...,x_m)=(x_1,...,x_m,0,0,0,...,x_n=0)$$ What is this type of function called?
3
votes
2answers
33 views

If $x \geq C$, where $C > 0$ is a constant, then what is the least upper bound for $\dfrac{2x}{x + 1}$?

The title says it all. Since $$f(x) = \dfrac{2x}{x + 1} = 2\left(1 - \dfrac{1}{x + 1}\right),$$ then because $x \geq C$ where $C > 0$, an upper bound is given by $$\dfrac{2x}{x + 1} < 2.$$ ...
1
vote
0answers
29 views

Finding value of definite integral - $f(f(x)) = x$ [duplicate]

Question: If $f(x)$ is such a function so that $f(f(x)) = x$ and $f(0) = 1$ then find the value of: $$ \int_0^1(x+f(x))^{2016}dx$$ I have no clue where to start this question. A small hint ...
1
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0answers
25 views

Understanding a theorem about double series

In baby Rudin, there is a theorem stated that: Given a double series $\{a_{ij}\},i=1,2,3,...,j=1,2,3,...,$ suppose that $$\sum_{j=1}^{\infty}|a_{ij}|=b_{i}$$ and $\sum b_i$ converges. Then ...
1
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0answers
17 views

Find Range of Domain and Co-Domain

Good Afternoon, I was wondering if someone could explain to me how to find the range of a domain and co-domain? For example, if A represents my domain and B represents my co-domain: $A = \{q,w,e\}$ ...
1
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2answers
31 views

Can the unit interval map bijectively to a region?

The Hilbert Curve shows that there exists a surjection from the unit interval to the unit square. I was wondering, does there exist a bijection from the unit interval to the unit square?
0
votes
1answer
31 views

How to derive relationship between two functions

I have two functions: $f(x) = x^2 + 200$ $g(x) = (x + 8)^2$ I am interested in the relationship between the two functions in the region between the two minimums (from x = -8 to x = 0), which ...
0
votes
1answer
26 views

Stuck relating the input to output (Transfer function)

i want to find the transfer function of a differential equation (given below) $\ddot\theta = a [ ([b\times Xin] - bk\dot\theta) - \ddot\theta] - c\phi $ (where $\phi$ and $\theta$ are time ...
0
votes
1answer
43 views

Discriminant of differential equation

I am aware of what the discriminant ($b^2-4ac$) means in relation to a function $f(x)$ when referring to the number of real roots. $$\begin{align} b^2-4ac&=0& &\longrightarrow ...
4
votes
2answers
43 views

Is the space of $\mathbb{R}\to\mathbb{R}$ more huge than the space of all discrete functions?

Assume that we have some general discrete function ${0,1,2,....n}\to\mathbb{R}$. For each number I have a real value. Let's infinite increase number $n$ (which is integer value), to approximate ...
4
votes
4answers
79 views

How to find the domain of this function?

f(x)= $\frac{(\sqrt{x}-\sqrt{x-1} )}{( \sqrt{x}+\sqrt{x-1} )}\;$ first off $\sqrt{x}$ is defined for: $$x > 0 \tag{1}$$ and $\sqrt{x-1}$ is defined for: $$x \ge 1 ...
0
votes
2answers
25 views

Reason for Fourier coefficients vanishing

I was computing the Fourier coefficient of the function: $$ F(t)=\left\{ \begin{array}{rl} F_0,&0<t<\pi,\\ -F_0,&\pi<t<2\pi, \end{array} \right. $$ with $F(t+2\pi)=F(t)$. Since ...
0
votes
1answer
41 views

Every irrational number is the limit of a sequence of rational numbers [duplicate]

How can I use this fact to show that f(y)=y for every real number y, given that f is continuous
1
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2answers
28 views

How do I solve for the derivative using quotient rule

How do I solve for $f'(x)$ when $f(x)=\frac{-e^x\sin x}{\cos x}$? Please show me the steps you took, I myself have spent about an hour on this :(
2
votes
1answer
36 views

Finding Conjugate harmonic of $u = \frac{1}{2} ln(x^2 + y^2)$

this is a nice community; I've been facing a hard time answering this question, a detailed answer would be splendid. $u = \frac{1}{2} ln(x^2 + y^2)$ find conjugate harmonic, and harmonic function. ...
1
vote
1answer
22 views

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function …

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function $f: A \to \Bbb N$ such that $\forall a, b \in A$ the following is true: $$a \mathcal R b \iff f(a) = ...