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

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0
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0answers
21 views

From wind speed to Watt

I have this graph below which show how much effect a windspeed make i Watt. My question is. How can I convert wind speed into Watt without reading the graph.
0
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0answers
26 views

The functional $\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$

Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$. Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$. Let $G\triangleq\{\varphi\in ...
1
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1answer
39 views

Finding the inverse of a non linear function

Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the diffeomorphism given by $$F(x,y)=(y+\sin x, x) $$ Find $F^{-1}$. I know that the answer is $F^{-1}(x,y)=(y,x-\sin y$), this can be shown to be true by ...
-1
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2answers
50 views

What is the difference between a function and a formula?

I think that the difference is that the domain and codomain are part of a function definition, whereas a formula is just a relationship between variables, with no particular input set specified. ...
0
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2answers
41 views

How to convert a discrete function to a continuous function

I was wondering because of this: Trick to find if number is composite or prime Is there any formal method to convert a discrete function to a continuous function. For example take $n!$, how was the ...
6
votes
2answers
133 views

A continuous function such that $f(x)=(f(x))^2$ for all $x$ is constant

Let $f(x)=(f(x))^2$ that is continuous for every $x \in\mathbb R$. Prove using the intermediate value theorem that this function is constant. I noticed that the $f(x)$ could only be equal ...
0
votes
1answer
24 views

Prove that a $f(c)=c$ given an interval where the function is continuous over

If $f(x)$ is continuous over the interval $[a,b]$ $a,b \in R$ $ a<b$ such that $f(a), f(b)$ also belong to the interval $[a,b]$ Prove that there exists some value $c$ in $[a,b]$ such that ...
1
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1answer
35 views

Does this variation of Jensen's inequality hold?

The original Jensen's inequality in probability theory is generally stated in the following form: if $X$ is a random variable and $f$ is a convex function, then $f \left(\mathbb{E}[X]\right) \leq ...
0
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1answer
38 views

For which constants $a$ function is continuous

Let $f(x)= \lfloor x \rfloor \cdot\cos{(a\cdot x)}$ where $x\in \mathbb{R}$. Find for which real constants $a$ function is continuous. We know function $ \lfloor x \rfloor$ is continuous apart from ...
0
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1answer
17 views

Prove that a function has 2 solution and find one solution using the bisection method

$ln(x^2+2x+\frac{1}{2})=x$ Prove that this equation has 2 solution over the interval $[0,10]$ Find the two first digits of one of the solution using the bisection method. I started with defining ...
2
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1answer
53 views

Cardinality of $M$ , $M = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 2x + y \in \mathbb{N}$ and $x - 2y \in \mathbb{N} \}$

I posted a very similar problem just yesterday, and yet I'm still struggling understanding these kinds of problems, so if anyone could suggest some kind of material I could read/watch to understand ...
3
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2answers
114 views

Find the function

Find function $f: \mathbb{R} \to \mathbb{R}$ which has limit only at $0$ and $1$ I think function $f(x)=x(x-1)$ when $x\in \mathbb{Q}$ and $f(x)=-x(x-1)$ when $x \notin \mathbb{Q}$ satisfy ...
2
votes
1answer
44 views

General Solution to functional equation

I was wondering how to derive the solution to $$ \frac{f(x + (1-2x)) - f(x)}{1-2x} = f(x)$$ Which can be simplified to $$\frac{f(1-x) - f(x)}{1-2x} = f(x)$$ One idea is as follows. Consider the ...
2
votes
3answers
48 views

Question about the graph of the square root function [duplicate]

I know this question may be stupid but I've been studying for my test tomorrow and I'm so frustrated, I can't figure this one out. if we have a square root function like this: $y = \sqrt{x}$ wouldn't ...
3
votes
0answers
76 views

Calculate $\sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \ldots $

$$ \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}+ \frac{1}{2}\sqrt{\frac{1}{2}}}} \times\ldots$$ I already know ...
1
vote
1answer
28 views

Line passing trough two points

Theroem: Given two points $A$ and $B$ with coordinates $(x_A;y_A)$ and $(x_B;y_B)$ then the equation of the line passing trough both $A$ and $B$ is $$y= y_{A/B}+\frac{\Delta y}{\Delta ...
2
votes
1answer
33 views

Cardinality of $L$, $L = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x + y = 5\}$

Like the title says, $L = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x + y = 5\}$ I need to find the cardinality of L, I have an idea of an answer, but I don't know what function I need to ...
1
vote
3answers
124 views

Demonstration of $\int_{a}^b f(x) \,dx= 0 \Rightarrow f(x)\equiv0 $ [closed]

Good morning, Can you give me a help to demonstrate this proposition: $f$ is a continuous and not negative function on the interval $[a,b] \ a,b \in \Re $, Demonstrate: $$\int_{a}^b f(x) \,dx= 0 ...
1
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1answer
25 views

Continuity of a function in a mixed discrete-connected domain

Consider the following function $f: \mathbb{R} \times \left\{0,1\right\} \rightarrow \mathbb{R}$ $$ f(x,y)=\begin{cases}x, & (y=0) \\ k, & (y=1) \end{cases} $$ where $k$ is some real ...
0
votes
2answers
69 views

Why does an $n$th degree polynomial have at most $n-1$ turning points?

How can one explain that polynomial of degree $n$ can have up to $n-1$ turning points and $n$ intersections with the $x$-axis? If it is easier to explain, why can't a cubic function have three or ...
0
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1answer
24 views

Lebesgue integral, integer part x

$$ \int_{0}^{\infty} 10^{-2[x]} dx $$ How to solve it? is the Lebesgue integral. I drew a graph, it is piecewise continuous. Sum of this function will converge. But I can not understand how it all ...
1
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0answers
32 views

Constructing set of functions that give a good basis after a certain integral

I need a set of functions that can be used as a basis after a specific integration. In other words: I integrate a set of functions enumerated by a parameter $k$ where the integral depends on another ...
16
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2answers
305 views

$f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$.

Let $f\in\mathcal{C}^2(\Bbb{R},\Bbb{R})$ be a positive function such that $f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ does it implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$? ...
0
votes
2answers
51 views

To show set is closed and bounded in $\mathbb{R}$

I am having problem with this question , kindly please help me with this , Let $$S = \{x : x^{6} -x^{5} \leq 100\}$$ And $$T =\{x^{2} - 2x : x \in ( 0, \infty)\}$$ Then I have to show that set $S ...
0
votes
2answers
62 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
0
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0answers
38 views

finding formula from table

I'm implementing a sudoku solver using human way algorithm. Which have 3 constraint, different number ini row, cell and box. I googled and I got ...
4
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3answers
74 views

what is meant by $ f ∈ C^{2}[a, b] ?$

What is the meaning of $ f ∈ C^{2}[a, b] ?$ I think it says that $f$ is twice differential on $[a,b]$, isn't it?
0
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0answers
41 views

Algorithm-generating algorithm

Is there an algorithm that can create other algorithms based on any number of arguments? For example, a way to determine a function $ f (x) $ from a given input and a given output? I.e. if $ f (2)=4 $ ...
3
votes
1answer
92 views

Is $f(x)=0$ a polynomial function?

Is $f(x)=0$ a polynomial function? we know that constant functions are polynomials of degree zero But, does $f(x)=0$ follow this definition?
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3answers
47 views

Linear independence related with functions

Good day ! I don't understand the following problem: "Prove that the three functions $x^2,\cos{x},e^x$ are linearly independent" So I think so I have to prove that the linear combination: $a ...
7
votes
2answers
147 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
3
votes
2answers
50 views

Function with a set number of pre-images

Let $A,B\subset\mathbb{R},n\ge2$. Let $f:A\to B$ (not necessarily continuous) such that $\forall a\in A,f^{-1}(a)$ is a tuple of $n$ elements. I know that if $f$ in continuous, for $A=B=\mathbb{R}$ ...
1
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1answer
60 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
0
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1answer
38 views

Prove that a function is total, surjective, injective and find its domain of definition

Let $D = \{1,2,3,4,5,6,7,8,9,10\}$ Let $f:P(N) \to P(N)$, $f(B) = B \triangle D$ I said that the image of this function is: $P(N)$, is that right? It's pretty clear that this function is total ...
1
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1answer
23 views

Examples of functions with properties

I'm looking for example of functions defined on $[0,+\infty)$ with the following properties: 1) continuous, twice differentiable 2) $f(0)=0$, $\lim_{x\to+\infty} f(x)=1/3$ 3) $f^{\prime}>0$, ...
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6answers
111 views

To show that $f (x) = | \cos x | + |\sin x |$ is not one one and onto and not differentiable

Let $f : \mathbb{R} \longrightarrow [0,2]$ be defined by $f (x) = | \cos x | + |\sin x |$. I need to show that $f$ is not one one and onto. I have only intuitive idea that $\cos x$ is even function so ...
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votes
3answers
139 views

20 / 5 (2 * 2) can equal multiple answers? [duplicate]

20 / 5 (2 * 2)=1 20 / 5 (2 * 2)=16 It states both answers are equally correct? Is this correct? If so why? If not why? Could this be a matter of perception in the way the person reads the problem and ...
1
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1answer
32 views

The alternating sum of primes defines an injection

Define $\displaystyle\alpha(n)=\sum^n_{k=1}(-1)^{n-k}p_k$, where $p_k$ is the $k$:th prime. Show that $\alpha$ is an injection $\mathbb Z_+\to\mathbb Z_+$. It's easy to see while considering sums as ...
6
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5answers
2k views

Formula to map 1 to 5… to 5 to 1 [closed]

Is there a simple formula to describe the following mapping? (I'm guessing it would be based on reciprocals.) $1 \mapsto 5$ $2 \mapsto 4$ $3 \mapsto 3$ $4 \mapsto 2$ $5 \mapsto 1$ (Really, I'd ...
3
votes
2answers
55 views

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$ I don't understand, how can we pass two things to a function? Can somebody explain what is this function, please?
1
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3answers
95 views

function such that $f(x\cdot t)=f(x)g(t)$

Let $E$ be the set of tuples of continuous functions $f,g:\mathbb{R}^*_+\rightarrow\mathbb{R}$ s.t. $f,g$ are never $0$ and $\forall x,t>0,f(x\cdot t)=f(x)g(t)$. I need to show that ...
0
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1answer
92 views

proof for zero function

I am given the following: Let $f$ be a real function, which itself and all its derivatives at $0$ are $0$. Assume there exists $b>0$ such that for all real $x$ and all natural $n$: $|f^{n}(x)|\leq ...
0
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1answer
34 views

Looking for another special kind of injective function

Relating to this Looking for a special kind of injective function Does there exist an injective function $f:\mathbb R→\mathbb R$ such that for every $c∈\mathbb R$ , there is a real sequence ...
2
votes
1answer
62 views

Is there an unbounded integrable function with integrable derivative in $(0,1)$?

I wonder if there is a differentiable unbounded function $f\in L^1(0,1)$ with $f'\in L^1(0,1)$. The elementary examples $x^\alpha$ or $\log x$ suggest that my question should be answered negatively. ...
0
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3answers
58 views

Sum of functions bounded between 0 and 1?

Let $y \in \mathbb{R}$ and $x \in \mathbb{R}$. I'm looking for two functions $f,g$ such that $$ f(x)+g(y) \in [0,1] $$ Do they exist? In positive case, do you have suggestions for what $f$ and $g$ ...
1
vote
2answers
41 views

To find the number of zeroes

P1 - Given $f (x) = x^ {3} + ax^ {2} +6x -1$ has critical point at $x=-2$, then how many real solutions has $f (x) =0$? MY Attempt regarding is that first i have found value of a which is 9\2 by ...
0
votes
2answers
52 views

X<5,Y<5 (clear)..but what if X<5, Y-X>10

I'm trying to construct geometric representation of the following: X<5, Y<5 (that is clear, it will be the area (square) with the corners on the 5s on X and Y axes. But I am clueless how to ...
5
votes
1answer
101 views

Closed form for sine graphic rotated by 45 degrees?

Is there a non-parametric closed form for a function looking like a sine rotated 45 degrees? I have encountered also a similar question but it asks for a function resembling the rotated sine, but not ...
1
vote
1answer
54 views

One-to-one functions of 2 variables

Are there any one-to-one functions of 2 variables? For each of the following prove or disprove whether there is a one-to-one function $f$ of 2 variables: $f$ is from $\Bbb{N}^2$ to $\Bbb{N}$ $f$ is ...
7
votes
6answers
299 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...