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

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0
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1answer
176 views

Conditions for unimodality of a sum increasing and decreasing functions

Suppose we have $f(x,y)$ and $g(x,y)$ where $f$ is increasing in both $x$ and $y$ and $g$ is decreasing in both $x$ and $y$. Are there any simple conditions for $f$ and $g$ so that $h(x,y) = ...
0
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1answer
37 views

Are those functions surjective, how would you prove it

Consider the functions $\text{add}:\mathbb{A}\to \mathbb{Z}$, such that $\mathbb{A}$ is a finite subset of $\mathbb{Z}$, and $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ ...
1
vote
2answers
72 views

Can a formula for this function be discerned?

I have access to a function, but I do not know the underlying algorithm. I'm attempting to guess it. I can feed it inputs and examine the outputs. I'll define it as $v(t,(x,y))$ where $t \in ...
1
vote
2answers
152 views

What is the difference between social choice function and social welfare function?

I am trying to understand a shard proof on Arrow's Impossibility Theorem and Gibbard-Satterthwaite Theorem. I stumbled upon these 2 different functions, and I cannot understand the difference between ...
0
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0answers
94 views

A question on convex/concave envelope

Suppose that we have a locally linear function y = f(x), i.e., it consists of several line segments. The question is: is there a numerical method that allow me to ...
2
votes
1answer
45 views

If $f(\bigcap_{\alpha\in{I}}A_\alpha)=\bigcap_{\alpha\in{I}}f(A_\alpha)$ then $f$ is an injection?

I know that if $f$ is an injection then $f(\bigcap_{\alpha\in{I}}A_\alpha)=\bigcap_{\alpha\in{I}}f(A_\alpha)$ but, the inverse is true?. I mean If ...
3
votes
0answers
45 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
1
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1answer
285 views

Undefined function

If we have an element of the set of real numbers i.e. '-3' and we have the square-root function i.e. $f(x) = \sqrt x$, then $f(-3) = \sqrt {-3}$ is undefined and the ordered pair $(-3, \sqrt {-3})$ is ...
1
vote
1answer
83 views

Supset proof of invertible functions

I'm struggling with the following question: $M$ is a non-empty set, $G$ is the set of invertible functions $f: M \rightarrow M$. Let $x \in M$ and $G'=\lbrace f \in G: f(x)=x \rbrace$. Now I have to ...
1
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1answer
67 views

Definition of square function

If we have two sets, the set of natural numbers and the set of integers and we relate each member of $\Bbb N $ to its squared value in $\Bbb Z$ then $f(x) = x^2$ and $\Bbb N $ is the domain of ...
0
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2answers
37 views

Defined amount and value amount of a function

What is the defined amount and value amount for this function: $$f(x)=\sqrt{(x+7)(1-x)}?$$ The defined amount is all the x-values the function can be and the value amount is all the y-values the ...
1
vote
1answer
148 views

How to find $\frac{d^2x}{dy^2}$

How to find $\dfrac{d^2x}{dy^2}$ in terms of $\dfrac{d^2y}{dx^2}$ and $\dfrac{dy}{dx}$ for any implicit function which is twice differentiable w.r.t. both x and y. I tried but we can't write ...
0
votes
1answer
254 views

Proving that $C$ is a subset of $f^{-1}[f(C)]$

More homework help. Given the function $f:A \to B$. Let $C$ be a subset of $A$ and let $D$ be a subset of $B$. Prove that: $C$ is a subset of $f^{-1}[f(C)]$ So I have to show that every element ...
1
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0answers
78 views

Smearing a line function into 3d space with Gaussion function

I want to smear a function defined along a straight line into 3d space with 2d and 3d Gaussion function, respectively. For example, in cylindrical coordinates $(r, \theta, z)$, $f(r)$ is a function ...
3
votes
1answer
35 views

Beginning Proof Question concerning Functions

So my class has been given the task to find functions $f$ and $g$,both from R to R such that: $f+g$ is differentiable and either $f'(0)$ dne, $g'(0)$ dne or both. I'm starting to believe, or at least ...
1
vote
2answers
49 views

Real function of one variable

A real function of one variable is a set $f$ of ordered pairs of real numbers such that for every real number $'a'$ one of the following two things happen: (i) There is exactly one real number ...
2
votes
3answers
86 views

Vertical line test

A vertical line crossing the x-axis at a point $a$ will meet the set in exactly one point $(a, b)$ if $f(a)$ is defined, and $f(a) = b$. If the vertical line meets the set of points in two points ...
2
votes
0answers
92 views

Upper bound for linear function

What may be more surprising is that when $a>0$, any linear function $an +b$ is $\mathcal{O}(n^2)$ which is easily verified by taking $c = a + |b|$ and $n_o = \max (\frac{-b}{a}, 1)$. $$an + b ...
1
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2answers
78 views

Set of periods of a real-valued function

Let $~f : E \rightarrow \mathbb{R}, ~ E \subset \mathbb{R}~~$ be a periodic fucntion and $ S = \{ T \in \mathbb{R} ~ : ~ \forall x\in \mathbb{R} ~~ f(x+T) = f(x) \} $ be the set of all periods of ...
2
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1answer
169 views

Cross Product for functions

So functions are just uncountabley-infinite dimensional vectors, and as such there's a nice generalization of the inner product between two functions (the integral of their product). Is their a ...
2
votes
2answers
59 views

How to find the range of $y = \frac{e^{x}}{1 + [x]}$

I am having trouble in finding the range of this function:- $y = \frac{e^{x}}{1 + [x]}$ Additional details are:- $x \geq0$ and [x] is the Greatest Integer function of x.
5
votes
6answers
737 views

What does the notation $f\colon A\to B$ mean?

I've been doing an online course in discrete mathematics, and the notation $f\colon A\to B$ has come up a few times, and it has not been explained what it means. I tried searching for it on Google, ...
3
votes
3answers
180 views

Strictly increasing function on positive integers giving value between $100$ and $200$

I'm looking for some sort of function $f$ that can take any integer $n>0$ and give a real number $100 \le m \lt 200$ such that if $a \lt b$ then $f(a) \lt f(b)$. How can I do that? I'm a programmer ...
9
votes
3answers
301 views

Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous

Solve the functional equation $$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$ with $f : [0,\infty) \to \mathbb R$ continuous. I can't manage to get this one ...
0
votes
2answers
55 views

Finding a function from the value of an inverse fuction

The function $f(x) = k(2 - x - x^3)$ has an inverse, and $f^{-1}(3) = -2$. Find $k$. I tried setting $f(x)$ equal to $3$ and plugging $-2$ into $x$ and I ended up with $3 = k(12)$. I'm not sure ...
2
votes
1answer
110 views

Derivative of a function equals the reciprocal of that function

I need to solve for a function that satisfies the following condition: $$f'(x)f(x)=1.\tag{1}$$ The solution suggests that we guess and verify. The guess is $$f(x)=Cx^k,$$ which implies that ...
1
vote
3answers
60 views

Ease numbers in a certain range

I'm trying to find an easing function taking in values from 0 to π/4 and outputting values in the same range, which starts ...
1
vote
2answers
62 views

Is there a name for a relationship like idempotence between two functions?

If $f(f(x)) = f(x) \quad \forall \space x$ then $f$ is idempotent. If $g(f(x)) = f(x) \quad \forall \space x$ then is there a term to describe the relationship between $g$ and $f$?
0
votes
1answer
56 views

Writing a formula for a spread sheet that can solve for a given value

For instance if given values are $f(1.1) = 1$, $f(1.9) = 2$, $f(2.7) = 3$, $f(3.5) = 4$, $f(4.3) = 5$ and ever increasing by $0.8$ for each number, if I were to enter a value of $1.5$ the answer is ...
2
votes
1answer
68 views

What is the opposite of a lift?

If I have a function $p:\tilde X\to X$ and a function $f:Y\to X$ , then a function $\tilde f:Y\to\tilde X$ such that $p\circ\tilde f=f$ is called a lift of $f$ with respect to $p$. So a lift is just a ...
1
vote
1answer
115 views

Conjectures prove or disprove

Prove this conjecture by giving 3 examples or disprove it using 1 counterexample. Prove the function $f(x) = \sqrt{x-2}$, over the domain $1 \leq x \leq 6$ will give a real solution. Please help me ...
3
votes
1answer
2k views

Power Series representation of $\frac{1+x}{(1-x)^2}$

Can anyone work out how to do this problem, because I'm getting an answer that close to the answer in the back of the book, but mine is off by a + 1. What I do is, I first find a representation for ...
1
vote
0answers
176 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
2
votes
2answers
336 views

Formal Dirichlet-Bourbaki definition of function

What is the formal Dirichlet-Bourbaki definition of a function? I have come across this in this essay: http://www.k-12prep.math.ttu.edu/journal/contentknowledge/meel01/article.pdf on page 1. I know ...
2
votes
1answer
738 views

Sequence of continuous functions converges uniformly. Does it imply the limit function is continuous?

Let $f_n \in C[a; b]$, where $\{f_n\}$ converges uniformly to $f$. Is it true that $f \in C[a; b]$ too ? How do I prove or disprove it?
0
votes
1answer
95 views

Is the canonical projection of $G$ onto $G/\ker f$ a lift of $f$?

I'm not sure I understand what a lift is and why it is called that. If I understand the definition correctly, then if I have a group homomorphism $f:G\to H$, then the canonical projection $\pi:G\to ...
2
votes
4answers
110 views

$f(x - 1) + f(x − 2) $ and the sum of coeficients

If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$ and $f(x)= ax^2 + bx + c$ what would be the value of $a+b+c$? I was doing $f(x-1)+f(x-2)= f(x-3)$ then $f(x)$ ...
0
votes
1answer
64 views

Power and automatic differentiation

I'm programming a math application where users can define functions and the application evaluates it at a given point. I'm trying to develop the automatic differentiation so I have developed a type ...
0
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0answers
28 views

Is this equation on the right form?

Lets assume there is a list $l$, where its items are denoted as $[a_0,...,a_n]$ and where we only consider the first and last third without the elements in b/n and while doing it recursively until we ...
0
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0answers
60 views

Minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$. Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy: ...
3
votes
1answer
196 views

Counterexample for a non-measurable function?

I am struggling to solve an exercise in my measure theory book and any help for solving it would be appreciated: Let $(\Omega,\mathcal{A},\mu)$ be a measure space and let $f:\Omega \to \mathbb{R}$ ...
0
votes
3answers
72 views

Is the map $f: \mathbb{U}^2 \setminus \{(1,0)\} \to \mathbb{R}$ surjective? [closed]

Define $f: \mathbb{U}^2\setminus \{(1,0)\} \to \mathbb{R}$ by $f(x,y) = \dfrac{y}{2(x-1)}$. Is this map surjective?
2
votes
1answer
46 views

Finding domain of a function

Finding the domain of the function $$f(x)=\frac{3\sqrt{x}}{x^2-5x-14}$$ My working out First I factorise the denominator $$f(x)=\frac{3\sqrt{x}}{(x-7)(x+2)}$$ Therefore the Domain is bigger than ...
0
votes
1answer
71 views

Need Assistance Setting Up Equation(s)

I am in a discrete mathematics class for information technology. I need some help setting up the equation(s) for a particular problem. It has been difficult to get individual attention with my rather ...
4
votes
1answer
120 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
-1
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3answers
82 views

How to find the inverse of a function $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ defined by $f([a])=[7a]$

If $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ is a function defined by $f([a])=[7a]$, show that $f$ is one-to-one and onto, and find $f^{-1}$. I've got proof that the function is well defined, one to one, ...
0
votes
1answer
45 views

Mapping intervals exponentially

I have two intervals: $X=[A,B]$ and $X'=[C,D]$. If I'd like to map $X$ to $X'$, I usually use this equation: $$f(t)= \frac{D-C} {B-A} t + \frac{(BC - AD)} {B-A}$$ where $t$ is the time. However ...
2
votes
3answers
200 views

If a bounded function $f:\Bbb R\to \Bbb R $ and $\left|\,f(x)-f(y)\right|<\left|x-y\right|$ for $x\ne y$, then there is an $a$ s.t. $f(a)=a$.

If a bounded function $f:\Bbb R\to \Bbb R $ and $\left|\,f(x)-f(y)\right|<\left|x-y\right|$ for $x\ne y$, then there is an $a$ s.t. $f(a)=a$. What I know is $f$ should be uniformly ...
1
vote
1answer
724 views

factor-square property (FSP) of polynomials

The Factor Square Property (FSP) is the divisibility of the polynomial $f(x^2)$ by $f(x)$. Is $x^2+x+1$ the only FSP irreducible polynomial of degree $2$ ? Are there other linear polynomial besides ...
1
vote
1answer
92 views

Check convergence of $f_{n}(x)=x^{n}-x^{2n}=x^{n}(1-x^{n})$

Check convergence of $f_{n}(x)=x^{n}-x^{2n}$ where $x\in(0,1)$ Please verify my answer, I'm not sure I'm doing it correctly. Thanks in advance!