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

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Generating function from a set of arbitrary integer values

Is it possible to generate a function defined in domain $\mathbb{R}$ which curves through a set of points $(n, a_n)$ where the function's local minimum and maximum between $(x_i, y_i)$ and $(x_{i+1}, ...
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1answer
11 views

Which function is suitable to approximate a convex piecewise linear function

Im trying to fit a convex piecewise linear function into a smooth function. However I have no idea which kind of function is suitable? Can anyone give me some examples of function that is suitable ...
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3answers
55 views

Is there a purpose behind a function?

As I understand it, a function is a relation between two sets of numbers where as for every input value there is only assigned one output. Or for every x there is only one y. What I don't ...
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2answers
49 views

Prove that f is injective. (Is my solution correct?)

Let $f: R\to R$, and ($f\circ f\circ f)(x) = (f\circ f)(x) + x$, $x$ $\in \mathbb R.$ Prove that $f$ is injective. My Solution: Let $x_1, x_2\in \mathbb R.$ and $f(f(x_1)) = f(f(x_2)) = y$ ...
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1answer
12 views

On determining a function given a certain parametrization of a point

Imagine we have a parametrization of a particle in 2D space like this http://i.minus.com/iXL64EfdJe6w5.gif How do we go about finding an explicit way to express these functions ($f(x)$ and $g(x)$) ...
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1answer
30 views

$f(x)=2-|x-3|, 1\le x\le 5$ and for other values, $f(x)$ is obtained using the relation $f(5x)=kf(x)$ for $x\in R$. then…

Question: The maximum value of f(x) in $[5^4,5^5]$ for $k=2$ is? Also, if $$\lim_{x\to \infty}\int_1^xf(x)dx$$ is a finite number, find the exhaustive set of $k$. Attempt : For first part, ...
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1answer
43 views

Are custom named functions acceptable notation?

A custom name being, for example, my function name (MFN): $MFN(x) := ax + b$ As contrasted with: $\delta(x) := ax + b$ Questions: Is it permissible to name the function $MFN$ above? Or is this ...
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1answer
24 views

Tight bounds for Bowers array notation

This link http://googology.wikia.com/wiki/Array_notation shows the definition of bowers linear array notation and the approximation $$\{n,a+1,b+1,c+1,d+1,...\}\ \approx f^a_{...+\omega^2d+\omega ...
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3answers
35 views

If a function has an inverse then it is bijective?

I have some trouble finding the answer to this, can someone help me out: If I have a general function $f$ with domain $X$ and codomain $Y$, I know nothing about the function (injective, surjective). ...
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1answer
30 views

Find all functions f with the following two properties

Let $f(x): [0, +\infty)\mapsto \mathbb{R}$ be a function such that for one $k\in [0, +\infty)$: $$f^2(x)=k^2+x\cdot f(x+k) \quad \forall x\in \{\;[0, +\infty) : x\geq k\;\}\qquad (1)$$ and ...
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1answer
35 views

If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
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1answer
29 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
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1answer
26 views

Solving function notations. [on hold]

A gas station attracts customers by offering coupons worth a 0.05 discount for every $1.00 spent on gasoline. a. Complete the table. The equation that represents the relation, using the ...
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1answer
58 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [on hold]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
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0answers
37 views

What function is this? [on hold]

guys . What function do these pairs represent ? (x , y) : (-1 , -0.5) , (0 , 1.5) , (1 , -2) , ( 2 , +4)
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2answers
37 views

Solving functional equation gives incorrect function

Let $f:\mathbb{R} \to \mathbb{R}$ be a function which satisfies $e^xf(y)+e^yf(x)=2e^{x+y}-e^{x-y}$ for all real x and y. If I place $x=y$, I get $f(x)=e^x-\frac{1}{2}e^{-x}$ which does not satisfy the ...
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1answer
25 views

Partial vs. complete definition of a function

Suppose I define a function as $f(x,y)=2(x+y)$. Compare that definition to $f:\mathbb{R}^{2}\rightarrow\mathbb{R}, f(x,y)=2(x+y)$, which also gives the (co-)domain. Is there any standard way to refer ...
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2answers
57 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
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2answers
574 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
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3answers
162 views

Create a function where $f(1000)=1.99$ and where $f(400000)=0.49$

As per object I need to create a continuous and decreasing function with one variable where result is as above, but not only. Function should also be easy to modify to get $1<f(1000)<10$ and ...
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3answers
67 views

Consider the function f(x)=sin(x) in the interval x=[π/4,7π/4]. The number and location(s) of the local minima of this function are?

This is MCQ of a competitive exam(GATE), Answer is (d) given by GATE , and from other sources ,explanation is (b) somewhere and (d) somewhere , I am going with (b) as minimum at $270$, I have drawn ...
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0answers
30 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
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0answers
23 views

Hieroglyphic from Herschel to Babbage?

John Herschel sent a letter to Charles Babbage in which he included this hieroglyphic with the message "Interpret it, it contains a great discovery". Personally I have no clue what it could mean. ...
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1answer
26 views

Finding the functions for circular reflection and their inverted forms

I'm trying to solve this exercise: Show that the transformation of inversion in the unit circle is given analytically by the equations $$x'=\frac{x}{x^2+y^2}, y'=\frac{y}{x^2+y^2}$$Find the ...
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1answer
49 views

Example of $f:\mathbb{R}\to\mathbb{R}$ injective and bounded, but with inverse not bounded or injective.

I am trying to come up with an example of a bounded and injective function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}$ is not injective or bounded. What are examples that could apply in this ...
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5answers
404 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
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0answers
57 views

Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
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2answers
51 views

Prove that an equation has solution in R

Let $f:\mathbb R\to \mathbb R$, $x\in\mathbb R$ and $$f(x^2 + 3x + 1) = f^2(x) + 3f(x) + 1.$$ Prove that $f(x)=x$ has a solution $\in \mathbb R.$
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1answer
31 views

Non-linear system with functions

$f:\mathbb R\to \mathbb R$ monotonically increasing. Solve the system: $$\begin{cases} f(x) + x = f(y) + y\\ x^2 + xy + y^2 = 12\end{cases}$$ Since $f$ is monotonically increasing and ...
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1answer
22 views

Can functions within a matrix adjust its size?

I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix: ...
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1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
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1answer
48 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
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0answers
22 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
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3answers
32 views

How to Prove it is a Even function? [on hold]

Let $y=f(x)$ be a function such that for any real numbers $a$ and $b$, $f(a+b)+f(a-b)=2[f(a)+f(b)]$ Prove that $f(x)$ is an even function.
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2answers
55 views

function: bending the y=x line

My question has many relative questions but I didn't find anything exact to my needs. Let's take the function $f(x)=x$ with $x\in[0,100] $. I need to bend this and make it a curve. f will be a ...
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1answer
35 views

Show that the angle between $OP$ and the normal to the curve at $P$ satisfies the following

I'm struggling to answer the following question below I've already worked out the gradient to the curve at $P$, but I'm having difficulty answering the second part of the question. MY attempt is as ...
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0answers
48 views

How do you the roots of functions that are not quadratics?

I was asked to consider the equation $(x-3)(x+3)^2=c$ I have been asked to find the values of C in which the equation has: three distinct roots only one real root a double root and a single root ...
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1answer
67 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
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1answer
75 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\mathrm{d}t$

Suppose $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, \mathrm{d}t$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I ...
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4answers
78 views

Evaluate $ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$ [on hold]

How to evaluate limit of the following function at x=0 ? $$ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} } $$
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5answers
69 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
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5answers
98 views

Show that $f(x) = \log(x + \sqrt {x^2+1})$ is an odd function

I need to show that $f(x) = \log(x + \sqrt{x^2+1})$ is an odd function and from what I can understand from this question (found while searching): What is an odd function?, I have to show ...
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1answer
40 views

Prime Number Algorithm

function isPrime(n) { // If n is less than 2 or not an integer then by definition cannot be prime. if (n < 2) {return false} if (n != Math.round(n)) {return false} // Now assume that n is ...
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2answers
29 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
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0answers
16 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
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0answers
29 views

Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
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0answers
25 views

Functon fitting goes wrong

Let's say I got a some function (let's say it named $B_w$) and I make a curve deped on some parameters. As example ...
0
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2answers
43 views

Maximum value of $f(x)=\frac{x^2}{x^3+200}$ over natural numbers

This was a great problem I came across today.Just wanted to share it :-) $f$ is a function defined over the set of natural numbers(I mean the domain is natural numbers) by ...
5
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4answers
74 views

Asymptotic of Inverse Function

Suppose we choose a positive constant $c$ and let $f_c(x)=\frac12x^2+cx^{3/2}$. I would like to get an asymptotic estimate for the function $f_c^{-1}(x)$ as $x\rightarrow\infty$. I assume it will be ...
5
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1answer
57 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?