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

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0answers
35 views

Urgent assistance in determining the function for a set of data?

I apologize if I cannot help anybody with my question right now. It's simply that I don't know what type of function represents the data I have plotted in excel, and therefore I cannot specify so in ...
2
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1answer
42 views

Derivation of Running(Online) Variance's formula

I need to know the dimostration of Running Variance's formula: $$ \sigma_n^2 =\frac{(n-1)\sigma_{n-1}^{2}+(x_n-\overline{x}_{n-1})\cdot(x_n-\overline{x}_{n})}{n} $$
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5answers
103 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
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2answers
41 views

Proving monotonicity of functions

For functions $f,g: \mathbb{R} \to \mathbb{R}$ prove the following: 1) If $f$ and $g$ are monotonic going up so is $f+g$ 2) if $f$ and $g$ are monotonic going up so is $f \cdot g$ 3) if $f$ and $g$ ...
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1answer
53 views

Are these examples of functions $f:\mathbb{N}\to\mathbb{N}$? Or are those examples not well-defined?

I have three examples of candidate functions here, and I'm wondering if they really are functions, or if they are not well-defined. The function $f:\mathbb{N}\to\mathbb{N}$ where $f(n)$ is the ...
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3answers
67 views

A difficult equation containing exponent 2 and 3

I couldn't solve this equation: $$ \frac{2}{x^2} + \frac{2}{2x} - \frac{(x+1)^2}{x^3} = \frac{1}{27} $$ Do I have to multiply everything by $x^3$ and also the righthand side $1/27$? $1 \cdot x^3/27 ...
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1answer
32 views

Writing down a two-dimensional function corresponding to the solid one generates by rotating a 1D function about its axis of mirror symmetry

Say I have a 1D function like $f_x = cos(x)$ that has reflection symmetry with respect to some axis, here $x = 0$. Making $f_x$ a Gamma distribution with arbitrary parameters $(k,\theta)$, or a ...
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1answer
178 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
3
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2answers
236 views

Does the bijection of function sets imply bijection of sets?

Let $X,Y$ and $Z$ be sets. Is it true that if the set of functions from $Z$ to $X$ is in bijection to the set of functions from $Z$ to $Y$, then $X$ is in bijection to $Y$? Or are there any subtleties ...
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2answers
72 views

Is the collection of all functions between two sets a set?

Can we say "the set of all functions between two sets" as easily as we could say "the set of all real numbers", for example?
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1answer
41 views

Floor and Ceiling question

This was a homework question. I wasn't able to get far because I couldn't determine the properties of floor and ceiling functions. Any help would be awesome. $\def\lc{\left\lceil} ...
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1answer
34 views

How to simplify $(3\sqrt{x})^3$

Simplify: $(3\sqrt{x})^3$. Where should I begin? I have tried to take to whole thing to the 2/3 power but that didn't seem to work.
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0answers
53 views

Quick Question on showing a function is an inner product

I just have a quick question How come =p(1)q(1)+p(2)q(2) is an inner product but =p(1)q(1)+p(2)q(2)-p(3)q(3) is not?
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0answers
67 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
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2answers
44 views

Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
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4answers
138 views

Interval around a root of a function

I have a question that may seems stupid and obvious, but for me it's not. The question is the following: Edit: I've explained bad my question Suppose we have $f: \Bbb{R} \to \mathbb{R}$ a non ...
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1answer
63 views

Is there any interesting interpretation of the set of all functions between two sets?

Is there any way to interpret the set of all functions from a set $X$ to a set $Y$? There is an interpretation of it as the cartesian product of $X$-many copies of $Y$, but I am asking for a more ...
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2answers
319 views

Mathematical roses with $4n+2$ petals

In polar coordinates $(r, \theta)$, the equation $$r = \sin\left(a \theta\right)$$ gives a rose with $a$ petals if $a$ is odd, or $2a$ petals if $a$ is even. Thus, the number of petals generated for ...
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1answer
147 views

Intersection of strictly monotone, convex functions on interval with two given intersections

Given two functions $f,g:[0,1]\rightarrow[0,1]$ that are smooth, strictly convex, strictly monotone s.t. $f(1)=g(1)=0$, $g(0)=f(0)=1$ and $f(x_0)>g(x_0)$ holds for some point $x_0$, does it follow ...
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1answer
323 views

Prove there exists a positive integer $N$, such that for every integer $n\ge N$, $f(n)=n$

Let $f:\mathbb Z^+\to\mathbb Z^+$ be a function satisfying the following conditions: (i): For any positive integers $m$ and $n$, ...
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1answer
34 views

How to skew points on a line towards a given point on that line?

I'm looking for a function that will skew values towards an arbitrary given value (c) in one-dimensional space. For example, suppose a student response is scored (s) from 0 to 100. The threshold to ...
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1answer
65 views

Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
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2answers
43 views

Limit as $x$ approaches 2 is undefined?

Does following function have a limit if x approaches 2. Calculate what the limit is and motivate why if it is missing. $$ \frac{(x-2)^2}{(x-2)^3} =\frac{ 1 }{ x-2}. $$ I answered $\frac{1 }{ 0 }= 0 $ ...
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2answers
27 views

A tangent of the function $f(x)=x^3$

A function $f(x) = x^3$ is given. What is the function of any tangent to $f(x)$? Do I have to use derivation to solve this question? I have no clues.
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1answer
110 views

$n^n$ are the moments of a measure on the non-negative real line?

I would like to know if the numbers $1,1,2^2,3^3,\dots, n^n,\dots$ are the moments with respect some measure $\mu$ on $[0,+\infty)$, i.e., if there exists such a measure $\mu$ with $$n^n=\int_0^\infty ...
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2answers
59 views

Is this function inequality true?

Let $\lambda$ and $\lambda_L$ be the values of the function $f(x,y)$ at the optimum for problems \begin{align} \lambda=\max_{x}\min_{y}f(x,y) \end{align} \begin{align} ...
0
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1answer
17 views

asymtote and value of a function

A function is defined as $f(x) =ln(px + q ), x>1$ , it intersects $x$ axis at $(4,0)$ and has asymptote at $x=1$. We are asked to find value of $p$ and $q$. By substituting $(4,0)$ in the function ...
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2answers
92 views

How do I prove this bijection?

The number of $n$-digit binary numbers with exactly $k$ $1$s equals the number of $k$-subsets of $[n]$. I think i'm on the right track, but I'm confused on how to write how it's onto and 1-1. This ...
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1answer
22 views

Finding a function without knowing its structure but some conditions

I'm trying to find a function who meets this conditions but have no idea where to start. Just think it may be related to the function $Ca^{-\left(x-\mu\right)^2}$, If it really has this structure (or ...
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2answers
98 views

Example of right inverse which is not injective.

My book says Let $f:A\to B$ be a function where $A\neq\emptyset$. Then $f$ has a right inverse $g:B\to A$ iff it is surjective. It sounds like surjectivity is a sufficient condition for $f$ to ...
2
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1answer
42 views

Characterizing certain real functions

After reading this question, I became curious about these functions, $f: \mathbb{R} \to \mathbb{R}$, with the property that $f(a+b) = f(a) + f(b)$ and $f(ab) = f(a)f(b)$. Clearly the only constant ...
2
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5answers
193 views

Why surjectivity is defined by “for every $y$,there exist $x$ such that$ f(x)=y$” instead of “$x_1=x_2\Rightarrow f(x_1)=f(x_2)$”

I think injective and surjective is a dual concept. Injective: $f(x_1)=f(x_2) \Rightarrow x_1=x_2$ But the definition of surjective is so different. It's "for every $y$,there exist $x$ such that ...
2
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0answers
62 views

The cardinality of the function

I'm reading a book about cardinality of functions and while I was solving some problems of the book I saw this: Prove that the cardinality of a general function $f:K \to K$ is $n^n$, where $n$ is the ...
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4answers
90 views

conjugate function prove derivative

If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$. How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?
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2answers
68 views

Why does a left inverse not have to be surjective?

My book says: Assume $A\neq 0$ and let $f:A\to B$ be a function. Then $f$ has a left inverse if and only if it is injective. Let the left inverse of $f$ be $g$. Then we have $g\circ f=id_A$. ...
2
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2answers
49 views

Form for continuous decreasing function with two fixed points

I'm looking for a specific function $f(x)$ with the following properties: Continuous (no piecewise functions) and smoothly decreasing. $f(x)>0$ for $0\leq x < c$ $f(0)=1$ $f(c)=0$ where $c$ ...
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1answer
35 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
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1answer
17 views

Inequality Conditions

Let $h_{k}(x)>0$ and $\sum_{k=1}^{l}h_{k}(x)=1$ (Here, $h_{k}(x)$ are some continuous functions). Is the statement below correct or not? $f_{k}(x)<0$ when $g_{k}(x)=0$, $\forall x \neq 0$, ...
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2answers
187 views

An example of a function whose domain is the set of positive integers and range is the set of integers?

I was browsing through one of my old pre-calc books, and I feel a bit ashamed to say I can't think of a simple answer. It intuitively feels impossible, as there are half as many points in the domain ...
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2answers
38 views

Relationship between $f$ and $f^{-1}$ unclear

Say I have the function from $f:X \to Y$, $f(x) = 3$ when $x \ge 0, \ = 0$ when $x < 0$. $X$ and $Y$ both with the standard topology. Hence $f^{-1}(y) = [0, \infty)$ when $y = 3, = (-\infty, 0)$ ...
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1answer
32 views

introductory calculus - Help me find a function with a few properties

I was asked to find a function $f: \mathbb R^2 \to \mathbb R$ such that: 1) $f$ is continuous at $(0,0)$. 2) $f$ has directional derivatives at $(0,0)$ (does this mean $f$ is differentiable at ...
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2answers
74 views

How can I solve this equation $3x^5-x^2/2+x+1=0$

Maybe I am just too tired, but I dont know how to solve this?? Can you point me in right direction. Thanks!!!
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2answers
64 views

Inverse Trigonometric Functions problem

Hello mathematicians Today I am stuck with a rather simple problems. $$y_1=\sin(\frac{1}{2} \arccos(\frac{4}{5}))$$ $$y_2=\cos(\frac{1}{2}\arccos(x))$$ I am required to simplify both of the above ...
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0answers
22 views

An additive function with capability to retrieve individual components

Assume a set of points in a path $A,B,C,D$ and $E$. Starting from point $A$ assuming single direction there are 4 possible paths as $AB, ABC, ABCD, ABCDE$ with identifies 1 to 4 representing each ...
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1answer
146 views

Sigmoid function with separate control of derivative at 0 and sharpness of bend.

For a physical relationship, $f(x)$, I'm trying to model, i have a fairly good determination of some of the boundaries, such that $ f(0) = 0$ $f'(0) = B$ $ \lim_{x\to\infty} f = A $ So far, what I ...
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4answers
77 views

How is this not a 1-1 function?

Using calculus, how would you prove that $$y={{2\,x}\over{\left(x^3+1\right)}} $$ is not a 1-1 function?
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1answer
47 views

Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
2
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4answers
103 views

$\epsilon$-$\delta$ limit proof that $\lim_{n\to \infty}\frac{n^2-n+2}{3n^2+2n-4}=\frac{1}{3}$

I need to prove that $$\lim_{n\to \infty}\frac{n^2-n+2}{3n^2+2n-4}=\frac{1}{3}$$ using the epsilon definition. I'm having specific trouble understanding how to make it less than epsilon once I've ...
0
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1answer
21 views

Problem with convex function

In Papadimitriou book I found a problem. If I know that function $f$ is a convex function, and I have values $x_2,...,x_n$, is function $g(x_1) = f(x_1,x_2,...,x_n)$ also a convex function? I know ...
3
votes
1answer
610 views

What is the difference between Curve Fitting and Regression(Machine Learning)?

I know that Machine Learning regression algorithms try to find the function of the data. That is, if we have 1000 data points (x,y), to find a general continuous function that follows the trends of ...