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Function Investigation proof verification

Your answer is almost correct, $c_0$ must be $0$. Similar to Solve $f(m + n) = f(m) + f(n) + mn$ and $g(mn) = g(m)g(n)(m + n)$ for $f \colon \mathbb N \to \mathbb N$. Let $g(n)=6f(n)-2n^3$. Then \...
PatrickStewart's user avatar
0 votes

Prove function is in $\mathcal{O}(x\log(x))$

You don't need to prove that the limit is finite the limit can even not exist. However, you can easily prove that the function is bounded. Indeed, for $x\in \mathbb R_{>0}$, $n = \left\lceil\log_2(...
Kroki's user avatar
  • 12.9k
2 votes
Accepted

$f(f(x))=f(x)^{2013}$ continued…

Let $A$ be any set such that $x^{2013} \in A$ whenever $x \in A$. Examples are $[0,1], [0,\infty), [1,\infty) $etc. Let $f(x)=x^{2013}$ for $x \in A$ and let $f$ map elements of $A^{c}$ in any manner ...
geetha290krm's user avatar
0 votes

How is a morphism different from a function

In algebraic geometry, functions and morphisms are both important concepts, but they serve different purposes and have different definitions: Functions: In algebraic geometry, a function typically ...
Rowing0914's user avatar
1 vote
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Whether a function has an inverse and horizontal line test

In the context of math classes where you're told to use the horizontal line test, you're working with 2-dimensional graphs in the plane and "has an inverse" typically means you can define a ...
Pineapple Fish's user avatar
1 vote

Whether a function has an inverse and horizontal line test

You can always restrict the codomain to make a function surjective. So if $f:X \rightarrow Y$ is injective then the same function but with codomain $f(X)$, denote it $g:X \rightarrow f(X)$ where $g(x)=...
Samir's user avatar
  • 336
0 votes

Question about the PHP (The pigeonhole principle PHP)

An approach using contradiction goes as follows. Let there be no 2 such teams that have 1 player in common. Now, given any 2 teams, they may have either 2 common players or none. Observe that if a ...
abhinav kumar's user avatar
1 vote

What is the range of the function $\frac{\sqrt{x^2 + 4x + 3}}{x-5}$

The function is continuous. As $x$ approaches $5$ from the right, the numerator approaches $\sqrt{48}$ and the denominator approaches $0$ and stays positive, so the values are arbitrarily large and ...
Arturo Magidin's user avatar
0 votes

Solve functional equation $f(x+y) + f(x-y) = 2f(x)f(y)$

Partial answer Plugging in $y = 0$ gives us $2f(x) = 2f(x)f(0)$, so $f(0) = 1$. ($f(x) = 0$ would also be a solution, but would violate property #2.) Plugging in $x = 0$ gives us $f(y) + f(-y) = 2f(0)...
Dan's user avatar
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5 votes
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Find all functions $f: \mathbb R \to \mathbb R$ satisfying $f(xf(y) + f(x+y)) = y(f(x)+1)+f(x)$

Note that $f(x) \equiv -1$ is also a solution of the functional equation $$ f(xf(y) + f(x+y)) = y(f(x)+1) + f(x) , \qquad \forall x,y \in \mathbb{R} \tag{1}\label{fe}$$ To examine other possibilities, ...
Sangchul Lee's user avatar
1 vote

Solve functional equation $f(x+y) + f(x-y) = 2f(x)f(y)$

(Too big for a comment), If we additionally assume that $f$ is $C^2(\mathbb{R})$, then this regularity condition uniquely determines the type of the function. Subtracting both sides by $2f(x)$ and ...
Sam's user avatar
  • 1,685
5 votes
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Solve functional equation $f(x+y) + f(x-y) = 2f(x)f(y)$

$2f(x) = f(x+0) + f(x-0) = 2f(x)f(0)$ for all $x$. If there exist any $w$ so that $f(w) \ne 0$ then $2f(w) =2f(w)f(0)$ and $f(0)=1$. And if $f(x)=0$ for all $x$ that violates property 2. So $f(0) = ...
fleablood's user avatar
  • 124k
2 votes

How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$

Divide the domain $D$ into two subdomains: $$D_1:0\leq x\leq\frac{2}{\sqrt 5},0\leq y\leq x;\qquad D_2:\frac{2}{\sqrt 5}\leq x\leq2,0\leq y\leq\frac{\sqrt{4-x^2}}{2},$$ then \begin{align*} \iint_{D} \...
Riemann's user avatar
  • 6,665
1 vote

My question is whether given function is periodic or not

The function is indeed periodic because $\mathbb{Q}$ is closed for the addition so for example $\forall x \in \mathbb{R}, x \in \mathbb{Q} \iff x+1\in \mathbb{Q}$. This means that $\forall x \in \...
Serge Ballesta's user avatar
0 votes

My question is whether given function is periodic or not

The period of a function is the smallest positive $r$ so that $f(x+r) = f(x)$ for all $x$. But some functions will have an infinite number of $r$s but none of them being a smallest positive $r$ so ...
fleablood's user avatar
  • 124k
0 votes

Difference of two greatest integer functions

This can be seen to follow from the more general property that $[x+n]=[x]+n$ for all real $x$ and integer $n$. Namely one sees that $[x]-[x-1]=[x]-([x]-1)=1$. This more general property can be derived ...
Etropy's user avatar
  • 46
1 vote

Difference of two greatest integer functions

A more formal proof could be derived from the intuition of the graph. By definition, $[x - 1]$ is the greatest integer less than or equal to $x-1$, so we have: $[x - 1] \leq x - 1$. Then adding $1$ to ...
27rabbit's user avatar
2 votes
Accepted

Difference of two greatest integer functions

Sure. Let $[x] = n$. That means $n \le x < n+1$. Subtract $1$ from each of the terms and that tells us $n-1 \le x-1 < n$. And that means that $[x-1]=n$. So $[x]-[x-1]=n - (n-1) = 1$. If ...
fleablood's user avatar
  • 124k
0 votes

Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

While the problem was solved above, here is a method that uses a stronger result (Darboux's theorem) but shows how to arrive systematically at the function used in the MVT. Suppose that there doesn't ...
Sarvesh Ravichandran Iyer's user avatar
3 votes
Accepted

Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

Let $$F(x)=e^{-x}\left(f(x)+\frac{1}{e-1}\right),\quad x\in[0,1],$$ then $F$ is continous on $[0,1]$ and differentiable on $(0,1)$, $$F(0)=F(1)=\frac{1}{e-1},\qquad F'(x)=e^{-x}\left(f'(x)-f(x)-\frac{...
Riemann's user avatar
  • 6,665
1 vote

What is an example of a linear function that maps a matrix to a scalar? What makes it a 'function'?

Your statement "$f$ is the vector $v$" is incorrect. I understand you want to emphasize how $f$ completely depends on $v$ (and you can denote it as $f_v$ as such), but $f$ is still a ...
SoftAl's user avatar
  • 11
2 votes
Accepted

How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$

Let's take a look at the domain $$\Omega=\{(x,y)\in\mathbb{R}^2\colon 0\leq y\leq x,\, x^2+4y^2\leq4\}$$ and notice that $$x^2+4y^2\leq4\iff \frac{x^2}{2^2}+y^2\leq 1$$ In red we have $\Omega$, in ...
Davide's user avatar
  • 331
0 votes

Basis functions that are one at a point and zero at others

The function $ \phi:\Bbb R\to [0,1] $ suggested by @SassatelliGiulio works: $$ \phi(x) = \max\left(0,\min\left(\frac{x-x_0+\delta}{\delta},\frac{x_0+\delta-x}{\delta}\right)\right) $$ Which leads to: $...
Megidd's user avatar
  • 221
1 vote
Accepted

Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

Here's a simple approach that does not rely on calculus. To see that $f(N)$ is positive, rewrite as $$f(N) = \frac{(N-2)(N-3)}{6N(N+1)},$$ which is $>0$ when $N>3$. To see that $f(N)$ is ...
RobPratt's user avatar
  • 44.9k
1 vote

Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

The function $f: \mathbb{N} \rightarrow \mathbb{R}$ defined as $f(N) := \frac{1+N^2}{N(N+1)}$ can have its domain extended to the positive reals; indeed for each $x>0$ define $f(x) := \frac{1+x^2}{...
Mike's user avatar
  • 20k
0 votes

Set theory - functions: Given that $|A|=|B|$, how to prove that $|A^C|=|B^C|$?

I did something like this: Let $\psi$:$A\longrightarrow B$ be a bijection since A and B are equinumerous.Then; define a function such that; $\phi$:$A^C\longrightarrow B^C$ as $\phi(f)=\psi$$\circ$$f$ ...
Elfryion's user avatar
0 votes

Analytic continuation of double factorial

$$(x)!!=\lim_{n\rightarrow\infty}\left(2n-\sin^{2}\left(\frac{\pi x}{2}\right)\right)^{\frac{x+\sin^{2}\left(\frac{\pi x}{2}\right)}{2}}\prod_{k=1}^{n}\frac{2k-\sin^{2}\left(\frac{\pi x}{2}\right)}{2k+...
Kamal Saleh's user avatar
  • 6,405
1 vote
Accepted

Proving this function is negative for all $x\in[0,\infty)$

It is in fact possible to prove the stricter result that $g(x)=f(x)-(c-1)$ is negative on its natural domain $D=(-\infty,1-ac)\cup(1-bc,\infty)$ (the argument of the natural logarithm must be positive)...
DarkLordOfPhysics's user avatar
1 vote

Can a function be defined as the union of two other functions?

Defining the union of 2 functions in that way is an oversimplification. Given 2 sets $X$ and $Y$, you can define a function $f:X\to Y$ as a relation $\mathcal{R}$ between $X$ and $Y$ verifying: $$\...
Serge Ballesta's user avatar
5 votes

Can a function be defined as the union of two other functions?

What you're doing is ok only if the functions have disjoint domains (or they happen to be equal on any domain overlap). This is not a very typical situation. In practice, any two functions you might ...
leftaroundabout's user avatar
29 votes
Accepted

Can a function be defined as the union of two other functions?

In the language of set theory, using the standard representation of function as sets of pairs, you are exactly right. Your $f$ and $g$ are both functions, as is $f \cup g$, and $(f \cup g)(2)=f(2) = 3$...
Rob Arthan's user avatar
  • 48.1k
0 votes

$g(f(x + y)) = f(x) + (x+y)g(y).$ Value of $𝑔(0) + 𝑔(1)+\dots+ 𝑔(2024)$?

$$g(f(x+y))=f(x)+(x+y)g(y)$$ Let $y=-x$ $$g(f(0))=f(x)$$ Now in original substitute $x=0$ and $x=1$ to get $$ g(f(y))=f(0)+yg(y)\\g(f(y+1))=f(1)+(y+1)g(y) $$ Subtract both to get $$ g(f(y+1))-g(f(y))=...
RandomGuy's user avatar
  • 1,114
1 vote

Finding the minimum value of function ${27𝑎^6}$ +$\frac{1}{(𝑎^2−𝑏^2−𝑐^2+2𝑏𝑐)}$

AM-GM generally works best when you get a constant on the other side $$ {1\over 4}(27a^6+{1\over 3a^2}+{1\over 3a^2}+{1\over 3a^2})\ge\sqrt[4]{1} $$ So $$27a^6+{1\over a^2}\ge4$$
RandomGuy's user avatar
  • 1,114
0 votes

What qualifies as a polynomial?

We would need to define a polynomial. A polynomial in terms of $x_1, x_2,...x_n$ is a particular type of mathematical expression with the following restriction: The $x_i$s are only added , subtracted ,...
Michael Ejercito's user avatar
1 vote
Accepted

Finding the minimum value of function ${27𝑎^6}$ +$\frac{1}{(𝑎^2−𝑏^2−𝑐^2+2𝑏𝑐)}$

You have to minimize the function $$f(a) = 27a^6 + \frac{1}{a^2}$$ with respect to $a$. For that, you can simply take the first derivative and equate it to $0$. Following these steps, you first get ...
jaumeap's user avatar
  • 34
1 vote

If $f$ is injective (resp. surjective), is $af$ injective (resp. surjective)?

Assume $f:\mathbb{R} \to \mathbb{R}$ is an injective function. Then $f(x) = f(y) \iff x = y$. Let $a \in \mathbb{R}-\{0\}$. Let's see that $af$ is also injective. If $af(x) = af(y)$, then $\frac{1}{a}...
ZAF's user avatar
  • 2,583
3 votes
Accepted

finding a tight scaling bound (in terms of the Big-O notation) of a function of an infinite sum of $1/n^2$.

The general method is to use the Euler-Maclaurin formula. In your case, you can notice that: $$ \frac1{n^2}\sim\frac1{n}-\frac1{n+1} $$ so: $$ \sum_{n=N}^\infty\frac1{n^2}\sim \sum_{n=N}^\infty\frac1{...
LPZ's user avatar
  • 2,590
2 votes

Can a discontinuous function be increasing or decreasing

Here's an example of a function $f$ which is discontinuous in every neighbourhood and strictly increasing everywhere. Let $r_0,r_1,r_2,...$ be an enumeration of the set $D$ of finite decimals between $...
John Bentin's user avatar
  • 18.2k
0 votes

Can a discontinuous function be increasing or decreasing

Let $f: \mathbb R \to \mathbb R, x \in \mathbb Q \implies f(x)=\tan x. x\in \mathbb Q^c \implies f(x)=0.$ It's only continuous at $x=0$. Not differentiable anywhere. If $x,y \in \mathbb Q \cap(-\pi/2,\...
TurlocTheRed's user avatar
  • 5,420
1 vote
Accepted

Some pattern in the gap of the extrema/root of a product using Stirling approximation

$x$ is a root of $f$ iff it makes one of the factors zero; that is, iff for some $n$ we have $g(x/n)=2$ where $g(t)=\frac{\Gamma(1+t)}{\sqrt{2\pi t}e^{-t}t^t}$. It happens that the function $g$ is ...
Gareth McCaughan's user avatar
1 vote

floor(x/y)=y graph

(see figures below) In such 2D questions, it is often rewarding to "take a step back" by considering curves as "level sets" (here at height $z=0$) of a surface, here with equation $...
Jean Marie's user avatar
  • 81.2k
2 votes

floor(x/y)=y graph

floor(x/y) is the integer $n$ satisfying: $$n \le \frac{x}{y} < n + 1$$ If $n = y$, then: $$y \le \frac{x}{y} < y + 1$$ If $y > 0$, then: $$y^2 \le x < ...
Dan's user avatar
  • 14.6k
5 votes
Accepted

floor(x/y)=y graph

The graph is a little misleading. It only includes the "flat" parts, not the "diagonal" segments connecting them. The complete graph is the union of the horizontal half-open ...
MPW's user avatar
  • 43.4k
0 votes

Does an infinite composition of functions converge? Can we determine it?

There should be many examples of an infinite composition also known as infinite 'iteration' that 'converges'. The only one I can think of right now is the Hahn Banach fixed point theorem: Theorem Let ...
César VB's user avatar
  • 435
3 votes

Does an infinite composition of functions converge? Can we determine it?

Yes. For a fixed value of $x$, say $x_0$, you obtain a sequence, $$\begin{align*} a_0 &= x_0\\ a_1 &= f(a_0)\\ a_2 &= f(a_1)\\ &\vdots\\ a_{n+1}&= f(a_n)\\ &\vdots \end{align*}$...
Arturo Magidin's user avatar
10 votes

Prove that for any function $f:\mathbb R\to\mathbb R$ there exist real numbers $x,y$, with $x\neq y$, such that $|f(x)-f(y)|\leq 1$.

Here's a proof. It may be what you are trying to say. The real line is the disjoint union of the countably many half open intervals $(n,n+1]$ of length $1$ (here $n$ is an integer). Since there are ...
Ethan Bolker's user avatar
  • 93.7k
0 votes

Finding solutions of modulus functions

The best way to solve this algebraically would probably be to split the functions up into piecewise functions. Let's say: $f(x)=|x-5|$ $g(x)=|3x-2|$ First, we need to know where the functions will ...
VV_721's user avatar
  • 66
1 vote
Accepted

How to visualize matrix functions?

Matrix functions are just functions on multiple variables. You can visualize them in the same way (or can't, as usually matrix functions are at least 8 dimensional, which is pretty hard to visualize ...
watertrainer's user avatar
0 votes

Non-computable function having computable values on a dense set of computable arguments

Noah Schweber seemed unsure about their approach to your second question, so I will give you the solution I found (which adapts Noah's work). Let $n\mapsto q_n$ denote any injective enumeration of the ...
Jade Vanadium's user avatar
1 vote
Accepted

Verifying the necessary and sufficient conditions for a function's inverse

No, they don't have to be inverses of each other, and they don't have to commute. An easy way to get counterexamples is to take two distinct involutions (functions equal to their own inverses). And ...
Arturo Magidin's user avatar

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