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

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What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
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1answer
46 views

What is the function $f$ such that $\sum_{k=0}^n f(k)=n^3$?

$$\begin{align*} 1 &\leadsto 1 \\ 1+3 &\leadsto 2^2 \\ 1+3+5 &\leadsto 3^2 \end{align*}$$ In general, if $f(x)=2x+1$, then $f(0)+f(1)+f(2)...f(n)=(n+1)^2$. Now, $$\begin{align*} 1 ...
1
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1answer
15 views

Find formula structure for a complex function

I am looking to find the function formula structure of a repeating function like the one in the image linked below.... Something that repeats indefinitely (like a sine wave) on the X-axis. Anybody ...
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0answers
27 views
1
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1answer
30 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$ , how can I prove that : $1)$ $h$ is an integrable function $2)$ $\int hdμ= \int fdμ- \int gdμ$
3
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2answers
177 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
0
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0answers
12 views

Piecewise Logistic Function [Satellite Data]

I am working with $16$-day MODIS EVI (satellite) data and I want to fit a Piece-wise Logistic Function through my $23$ EVI data values. The following formula is for the Piece-wise Logistic Function: ...
-1
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2answers
57 views

Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$. [on hold]

Let $f(x)$ be a polynomial in $x$ With integer coefficient. If for natural numbers $a,b,c$, $f(a)=b, f(b)=c, f(c)=a$ Prove that $a=b=c$.
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1answer
30 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
1
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2answers
41 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
1
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3answers
45 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ ...
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2answers
22 views

Can we say that a function is increasing/decreasing on some range if there's a vertical asymptote in that range?

The graph below shows the function $f(x)=\frac{e^x}{x-1}$ Can we say that the function is decreasing for all $x\le2$ (there's a local minimum at $x=2$) or do we have to take the asymptote at $x=1$ ...
2
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1answer
29 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...
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1answer
21 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
1
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0answers
63 views

Is there a name for the function of a semicircle?

Recently I've learned many different names for different types of functions... but I've been wondering, is there a name for this type of function? $\sqrt{x - x^2}$
3
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0answers
27 views

Technical name for an almost-monotonic function

I'm wondering if there’s a technical name or short phrase that describes a function that’s monotonic, subject to some uniformly bounded amount of backtracking. $\exists \epsilon \forall x , y : y \gt ...
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4answers
46 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
3
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3answers
66 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
0
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1answer
26 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
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0answers
20 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
-1
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1answer
21 views

Ratio known and Amount needed known

You have a a recipe for the perfect orange juice, it is: 26 parts Water to 1 Part Orange Juice (26:1). However overall you only want 20ml of mixed liquid (orange juice and water combined). What ...
3
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3answers
36 views

Finding periodic (trigonometric?) function given points

It's been a while since I've taken a math class. I need a couple functions for a program I'm working on. I can tell they involve trigonometry, but I can't figure out how to derive the function ...
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0answers
12 views

Function/algorithm to generate a random walk on a graph

I'm looking for a graph function or an algorithm that can generate a random fluctuating random walk that will eventually converge between the value of y = 0 and y = 1, more or less after a number of ...
1
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0answers
48 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
0
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1answer
35 views

Confused by one-to-one question, I think it's order incorrectly

I have this question and it seems a tad redundant If $A$ and $B$ are infinite sets, is it possible for there to be a 1-1 function from $A$ to $B$ and a 1-1 function from $B$ to $A$ without there ...
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4answers
66 views

Why is this function a bijection?

Consider the function below $$f:\mathbb{R^+} \to \mathbb{R^+}$$ given by $$f(x) = \sqrt{x}$$. Now it makes sense that the function is injective because $f(x) = f(y) \implies \sqrt{x} = \sqrt{y} ...
1
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1answer
110 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
0
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1answer
60 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
0
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1answer
47 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
0
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1answer
30 views

$f(x)=y$ while $g(y)=x$; Is it possible to find two not reverse functions that behave such at least for a given set of inputs and outputs?

I want to know if it is possible to program such a code that could determine two distinguish, not inverse, functions, say $f$ and $g$, that is true for the below statements at a given input and output ...
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0answers
23 views

Please write down exact output after the following statements are proceesed [on hold]

Please write down exact output after the following statements are proceesed ...
0
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2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?
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0answers
8 views

Value distribution into random variables

I want to distribute a large number into quantities of 100s, 50s, 25s, 5s, 1s Lets say, I'm selling apples in quantities of 100s, 50s .... 1. Based on the quantity there is different pricing. 1 apple ...
0
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1answer
29 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
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2answers
10 views

Is the sum of a unimodal and increasing function still unimodal?

There is no specific function. I would like to know if there is information on summation of a unimodal and increasing function? (Continuous functions)
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2answers
47 views

An injection from R × {0, 1} to R [on hold]

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
1
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0answers
11 views

A property of Quasiconvex functions.

Let f be a strictly quasiconvex differentiable function and Df denote its gradient. Is the following implication true? :"Whenever f(y) < f(x), we also have (Df(x))'(y - x) < 0" . Suppose that f ...
1
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0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
0
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0answers
9 views

how to show a concave function on discrete domain increases in x?

Define $g(x)=\frac{f(x)/x}{f(x)-f(x-1)}$ where x$\in$ $\mathbb{Z}$. Known that $f(x)$ has the concave extension in every consecutive $x$, i.e: $f(x+1)+f(x-1)-2f(x)<0$ holds $\forall x$. My question ...
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0answers
24 views

Consider the integral expression in $x$ [on hold]

$P(x)=x^3+x^3+ax+1$ where $a$ is a rational number. At $a=3$ the value $P(x)$ is a rational number for any $x$ which satisfies the equation $x^2+2x-2=0$, and in this case the value of $P$ is $12$.
5
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1answer
72 views

If $f(f(x))=x$ does that mean $f(x)$ equals its inverse?

Given any real function, if $f(f(x))=x$ does that mean $f(x)$ is its own inverse? I am confused since $f^{-1}(f(x))=x$ and this is a fact, so can we assume that $f(x)$ will equal $f^{-1}(x)$ by ...
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2answers
31 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
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0answers
11 views

Existence of subgradient of a quasiconvex function

Does a continuous quasiconvex function always have a subgradient? More strongly, is it true that if $f$ is a continuous quasiconvex function, then for each $x$ there is a vector $c$, such that for ...
1
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1answer
27 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
0
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0answers
27 views

Function that maps all vectors to the origin?

I need a function that will map any vector (and any point on that vector) in the cartesian plane to (0,0) using only addition and subtraction. Is this possible?
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0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
0
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1answer
37 views

Functions of modulus

How do I calculate the range of any modulus function? I know that if $x <2$ then it's expansion is negative and if $x>2$, it's expansion is negative, but will it help? Consider an example, $$f ...
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4answers
48 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
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2answers
54 views

What is n (Natural number) if the function has to have a limit not equal to zero or infinite? [on hold]

$$\lim_{x \to 0} \frac{(\tan(x))^n - x^n}{x^6}$$ What is n (Natural number) if the function has to have a limit not equal to zero or infinite?
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0answers
20 views

Mathematical formal expression of find “subfunction” in function

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...