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

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1answer
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Function Equivalent to a Constant Paradox

Say I define $z(x,y) = x^2+y = \text{constant}$ Then $\left(\dfrac{\partial z}{\partial x}\right)_{y} = 2x$ However, $\left(\dfrac{\partial \text{ constant}}{\partial x}\right)_y = 0$ Shouldn't ...
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2answers
72 views

Show that $f:\mathbb{R}-\{2\}\to\mathbb{R}-\{5\}$ with $f(x)=\frac{5x-1}{x-2}$ is bijective

Can anyone please help to explain the question and what actually $f: \mathbb{R} - \{2\}$ means ?? I know that bijection means one to one function and onto both. Any idea to start up with this ...
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1answer
25 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than 100cm. There is ...
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1answer
19 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
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1answer
34 views

Is there exist an additive but unbounded function?

I just learned that the function that is additive and bounded near $0$ on Real has the only form of $f(x)=cx$, where $c$ is a constant number. We say that a function $f$ is additive iff ...
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0answers
28 views

Functions commute with a given polynomial

Given a polynomial $f(x)\in \mathbb{C}[x]$,how to find(describe) functions(smooth or continuous or polynomial) that are commute(under composition) with $f(x)$? There are trvial ones :$x,f,f\circ ...
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2answers
65 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
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0answers
17 views

Notation Question with regard to functions

Let $f : N → N$ Let $E(f)$ be the function defined by $E(f)(n) = 2^{f(n)}$. Does $E(f)(n)$ mean $E(f(n))$? or $E(f)(n)$?
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2answers
27 views

Simplify a function with a square root as the numerator

How would I go about simplifying this: $$\frac{\sqrt{x^4 + 3x^2}}{x}$$ thanks!
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1answer
36 views

Is there a way to re-write $\min(a,b)$ in terms of an analytical function?

Is there a way to re-write $\min(a,b)$ in terms of an analytical function? Also, if not, is there a nice analytic function that is a tight upper bound? This question is related to this question.
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2answers
25 views

Big O notation - proof

Is it true that $O(k(n) + m(n))$ is equal to $O(\max\{k(n), m(n)\})$? In one of papers on computational complexity I've found the following statement: $$O(\log(n) + n(\log S + \log V )) = O(n(\log ...
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1answer
25 views

a problem on functional equations [duplicate]

if fuction is defined from $N to N$ then can we say that it is not continuous as it is not defined for all $x$??a simple statement ,but this is stopping me from giving the solution to a question..pls ...
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0answers
14 views

Rounding up approximations when using iterative methods

When using iterative methods I have read that as soon as you get two successive approximations that round to the same number of decimal places, then all further approximations will round to that ...
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0answers
25 views

Bijectivity of a function $f(i,j)=\frac{(i+j)(i+j+1)}{2}+j$

Define $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ by $f(i,j)=\frac{(i+j)(i+j+1)}{2}+j$. how can I prove that f is bijective help please! I should prove that f is surjective and ...
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1answer
45 views

Limit of a function containing square root.

Q. $\lim _{x\to 0}\frac{\left(\sqrt{1-cos2x}\right)}{x}$ We can write this function as $\lim _{x\to 0}\frac{\left(\sqrt{2sin^2x}\right)}{x}$. Algebraically we have ...
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0answers
12 views

How to find a function from a matrix?

Suppose I have a matrix like this: H2 N2 G2 H1 0 3 8 N1 2 4 7 G1 1 5 6 How would I find a nice function $f(x, y)$ that ...
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0answers
12 views

Algorithm that adds three numbers in an array that performs in O(n^2) time

Note that this question extends on this previous question. Given an array A, and a value called value. Does there exist three ...
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1answer
26 views

How to combine two functions into one continuous function so it can be integrated/differentiated?

I have a function like this : $$f(x)=\begin{cases}x \in)-\infty, 1)&,\;\;f(x)=x^2\\{}\\x\in(1,+\infty(&,\;\;f(x)=x^3\end{cases}\;\;\;\;\;\;$$ as you can see, the function as a whole is ...
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2answers
15 views

Designing a fast algorithm which adds three numbers in array

Given an array A, and a value called value. Does there exist three elements in A where their ...
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3answers
71 views

Use of $\mapsto$ and $\to$

I'm confused as to when one uses $\mapsto$ and when one uses $\to$. From what I understand, we use $\to$ when dealing with sets and $\mapsto$ when dealing with elements but I'm not entirely sure. ...
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3answers
285 views

Can I cancel out quotient function safely?

Can you actually cancel out the numerator and denominator? $$f(x) = \frac{x^3-8}{x-2}$$ This function is not defined at $x=2$ so the domain of it is "all real numbers except 2". $f(x)$ can be ...
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7answers
1k views

Function that is non-zero only at one point.

I am searching for, if there exists, a continuous function $f(x)$ such that $f(x) = 0$ for all values of $x$, with the exception of one point (say $\tilde x$) where $f(\tilde x)\neq0$.
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2answers
21 views

Help analyzing the time-complexity of my algorithm

So, (this is homework), we are given an array A, and we are asked to create a function where we return True/False if the array contains three elements which sum to a given value. To formalize: ...
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2answers
27 views

Meaning of -g(x) = g(-x) notation

If -g(x)=g(-x) for all values of x, then which of the following could be g(x)? What does the notation mean?
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0answers
42 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
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1answer
16 views

Series expansions and perturbation

My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this ...
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2answers
30 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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1answer
45 views

Elementary set problem

Let $A$ be a set and $f$ a function $f:A \to A$. A set $B$ is called "solid" in $A$ if and only if $f(B) \subseteq B$. Prove that a set $A$ is finite if and only if there is a function $f:A\to A$ ...
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0answers
13 views

Maximum of a function

I want to find the maximum of the following function with respect to $\omega$: $$ \max_\omega \left| \frac{1- e^{(-\alpha+j\omega) M}}{1-e^{-\alpha+j\omega }}\right|, \quad \omega \in[0, 2\pi[$$ I ...
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0answers
15 views

A proof problem about intergral equation's root

Several days ago,my junior asked me the following problem: Let $$F\left( w \right) = \frac{1}{T}\int_0^T {M{x_C}\left( t \right)\cos \left( {tw} \right)dt} - \frac{{\sin \left( {{T_s}w} ...
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0answers
33 views

The number of Balanced Boolean functions

Suppose we have n-variable Boolean function (BF) and we know that the weight of a Balanced BF is $2^{n-1}$. The total number of BFs are $2^{2^n}$, Affine BFs are $2^{n+1}$ and Linear BFs are $2^n$. In ...
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1answer
32 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
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1answer
35 views

Please explain the rules of differentiation?

I have an equation $f(x)=3\sqrt{\ x}$ and i have to find the derivative of the function f. What i have gotten so far is $3x^{-1/2}$, which then comes out to be ${3/2}x^{-3/2}$. I know the answer is ...
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1answer
16 views

Composite function domains

$f(x) = 1/x$ domain : all real numbers except $x=0$ $g(x) = \sqrt {x + 2}$ domain : $x$ is greater than or equal to $2$ I'm supposed to find the $f(g(x))$ and $g(f(x))$. This is simple ...
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1answer
57 views

Suppose all partial derivatives of $f$ exist at $x_0$; is $f$ continuous at $x_0$?

Consider $f : C \to \mathbb{R}$ with $C \subset \mathbb{R}^n$ being open: Suppose $f$ is differentiable at $\mathbf{x}_0 \in C$. Is $f$ continuous at $\mathbf{x}_0$? Why? Suppose all partial ...
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2answers
33 views

What is the 'formula' of a composite function?

Consider $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) = \frac{1}{x^2 +1}$ and $g: \mathbb{R} \to \mathbb{R}\times \mathbb{R}$ given by $g(x) = (3x, x^2)$. I was asked to find the 'formulas' of $f ...
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1answer
10 views

Not existance of one sided limit

I have a question regarding one sided limits of functions. Let's say that the function $f$ is defined in $(a,b)$. And let's say that we want to check the limit of $f$ when it approaches b from the ...
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1answer
31 views

Why are the following graphs discontinuous at $f(0)$ (epsilon-delta)

The caption for graph (f) is "Infinite jump". The caption for graph (h) is "Infinitely many infinite jumps". The graphs are meant to illustrate that we can pick arbitrarily small intervals around ...
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1answer
14 views

Period of two equal functions

I'm dealing with a problem here. We know that two functions are the same if they have the same domain and codomain. Let's say we have given the functions $f$ ang $g$ where ...
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2answers
25 views

How to calculate minimum value of a function?

How to calculate minimum value of a function? $min $ $f(x)=(x_{1}-2)^2 + (x_{2}-1)^2 $
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1answer
49 views

Prove that exponential functions grow faster than polynomial

I am asked to proof that being $r$ and $s$ two known fixed real numbers such that $r > 0$ and $s > 1$, there exists $n_0$ such that for every $n > n_0$ this happens: $n^r < ks^n$ where ...
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2answers
79 views

Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
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2answers
81 views

Finding example of a special type of continuous differentiable function

Give example of a continuous function (if exists) $f : [a,b]\to \mathbb R$ differentiable in $(a,b)$ such that $f(a)f(b) \ne 0$ , the set $A:=${ $x \in (a,b) : f(x)=0$ } is infinite but not an ...
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1answer
17 views

How can I prove this is surjective?

Let $f: \Bbb{R}- \{2\} \to\Bbb{R} - \{5\}$ be defined by $$f(x) =\frac{5x + 1}{x-2}$$ My understanding of proving surjections is that you must show $f(x) = y$ i.e that all elements in the domain ...
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29 views

Polynomial Functions/Remainder Theorem Challenge Problems

Please any help would be greatly appreciated! Refer to the photo for the question below. I know how to do it when given only one factor. How should I do it for this case? Thanks in advance for the ...
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2answers
97 views

Constructing an increasing function with prescribed values at three points

This should probably be very simple, but I'm just not very skilled in math :S. I want a function that takes one variable, x, ranging from 0-1. As the input approaches 0 so should the output. As the ...
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1answer
27 views

advanced algebra function word problem [duplicate]

can you please explain the answer and why you got it? the distance in feet $d(t)$ a dropped object falls in $t$ seconds is given by the function $d(t)=16t^2$. suppose you drop a ball from a height of ...
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1answer
35 views

prove or disprove the statements about functions [closed]

Is there a non-zero function $f:R\rightarrow R$ such that $f(x+y)=f(x)+f(y)$ for all $x,y \in R$, $f(xy)=f(x)f(y)$ for all $x,y \in R$, $f(x+y)=f(x)f(y)$ for all $x,y \in R$, $f(xy)=f(x)+f(y)$ for ...
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1answer
27 views

Is there a way to do this? Fixed deduction for x rounds where total = fixed amount

I am trying to calculate the reduction amount / step per round for the given: rounds = 1000 points = 80 starting at reward = 1 point So from round 1 which has a reward of 1 point deduct a fixed ...
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3answers
69 views

Set notation and mappings question

Good evening. I have a question. Suppose I have two sets, $A=\{1,2,3,4\}$ and $B=\{5,6\}$. I want to write the notation for a function that takes each element in $A$ and assigns to it a value in $B$. ...