**19.793 functions questions.**

Let us define a "scaling operator" $\hat{\Sigma}(\lambda)$ such that when it acts on any function $f(x)$, it gives $$\hat{\Sigma}(\lambda)f(x)=f(\lambda x).$$ Is it possible to represent the operator $...

Assuming I have $F(x): \mathbb{R} \to \mathbb{C}$ and I can estimate a first and second derivative at a point x. I'd like to use Newton's method to find maxima of $|F(x)|^2$, how can I do this? In ...

I want to find out the function of a graph that would show up as a straight line if plotted on the following lin-log scale.
I managed to do this for a different case where the function starts at $\{0;...

Let's say I have an integral $\int_0^1 f(t)dt$, being $f$ a continuous real function, and I add a continuous weight function $h(s,t)$ depending on a parameter $s\in[0,1)$, with the intention of ...

Let $f$ and $g$ be a pair of functions mapping reals to reals. It is common to use the point-free notation $f\circ g$ to describe the function $h$ defined by $h(x)$ = $f(g(x))$. By "point-free" I mean ...

I came across this very interesting question, which seems to be partially answered in a couple posts around here:
Let $f:[0,1]\rightarrow\mathbb{R}$ continuous such that $f(0)=f(1)$. Then for all $n&...

I actually want to know the proper usage of curly brackets and parentheses. I just used the function notation to express my question.
My question is that when using a function notation, we use a ...

Find all function $f: \Bbb{N}_0 \rightarrow \Bbb{N}_0 $ satisfying the equation $f(f(n) + f(n) = 2n +3k$ for all $n \in \Bbb{N}_0 $ where $k$ is a fixed natural number.
I have proceeded with ...

I know how incredibly stupid this sounds, but bear with me.
Let's take any random $x$, say $3$, and any random $-x$, say $-3$. Let's plug it into $x^2$. They will both give the same result!
I know ...

My textbook says that if $f (x)=\frac {g (x)}{h (x)} $ is a rational function of $x $ and $h(x)=0$ does not have any real root then $f (x) $ is continous and non monotonic.
$f (x) $ being continous ...

Does the function
$$f(x) = \frac{1}{1 + x}$$
have a recognizable name?
For example a related function with a recognizable name is the logistic function, defined by:
$$l(x) = \frac{1}{1 + e^{-x}...

I am studying for an exam in elementary set theory and I am not understanding the proof for this theorem:
For any three sets $A,B,C$:
$(A^B)^C \sim A^{(B \times C)}$
I know I need to find a ...

I am looking for a explicit expression of a $C^{\infty}$ function $h$ with $$h(x) = 1$$ for $x\in\mathbb{R}^{n}\setminus B_{1/4}$ and
$$h(x) = 0$$ for $x\in B_{1/8}$. Can someone help ?

I'm trying to show that the maps defined in https://math.stackexchange.com/a/413846/500094 are mutual inverses. I have problem with one direction:
$(g\circ f)(v)=g(d(\pi_X)_{(p,q)}v,d(\pi_Y)_{(p,q)}...

Is there a known function $f:\mathbb{R}\to\mathbb{R}$, such that:
The definition of $f$ does not contain the $!$ operator
The definition of $f$ does not contain the $\sum$ operator
The definition of $...

How to find the elements of $S$ where $S=\{x^6-x^5\leq 100, x\in\mathbb R\}$.
What elements are there in S, for that we need to find the range of S.Now how to solve this then.

I know it sounds really dumb, but is it true that $(f_1+f_2)\circ g=f_1\circ g+f_2\circ g$?
I know it must be really elementary, but I don't recall seeing this being proved (or defined) explicitly.

$F(x)$ satisfying the condition satisfying $f(x) +f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$
Answer is $f(x)=\frac{x+1}{x-1}$
I want to know the method

I'm trying to find simple mathematical proof that lengthy passwords are more secure than complex (with multiple symbols) ones. I don't want to get into the concept of entropy - it is not useful to me ...

Good evening,
I'd like to ask a very specific question about a method that my professor used within the following exercise :
Qualitative analysis of $ y' = \frac {(y+1) \cdot \sinh(x)}{(y+1) \cdot \...

Consider the following matrix function defined:
$$D_X(Y) = XYX^\top - \omega Y\omega^\top.$$
I have $D_\Gamma(L) = \frac{\partial}{\partial\vartheta}\Gamma$. Using the pseudo inverse of $D_\Gamma$, ...

I need to solve equation $$x^2+\ln x=0$$ I don't understand what my next step should be. I have tried to use $e$ and I have gotten $$e^{x^2}+x=0$$But that doesn't make any more sense to me than the ...

I am trying to model a phenomenon where the output depends linearly on a variable $x$, except when $x$ becomes large where its contribution is less important.
I would like to use a function looking ...

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...

I am a high school student of Mathematics. I have been working on this piece-wise function, trying to understand its properties:-
[N is the set of all natural numbers, W is the set of all whole ...

Find a function $f:\mathbb{N} \rightarrow \mathbb{R}$ satisfying $f(n+1)-f(n)>\frac{1}{12}$ and $\lim_{n \rightarrow + \infty} f(n)=\frac{2}{3}$
Is there any "simple" function satisfied ? (Like ...

I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$.
So I pose the following problem:
Given $a \in \...

I struggle to understand the following theorem (not the proof, I can't even validate it to be true). Note: I don't have a math background.
If S is not the empty set, then (f : T → V) is injective ...

A function $f \colon X \to Y$ is called surjective if, for every element $y \in Y$, there is an element $x \in X$ such that $f(x) = y$. There can be 1 such element in $X$, or 2, or a hundred - but you'...

The complete statement is: Let $(A,≤)$ be a well-ordered nonempty set and let $f$ be the function of $P(A)-${$Ø$} in $A$ defined for: $f(S) =$ first element of $S$.
(a) Prove that $f$ is surjective.
...

There are different scales used in mathematics - particularly when it comes to plot. Two major ones are linear and logarithmic. But I was wondering how can one grasp the whole set of numbers in one ...

$3f(x) + f(2-x) = (x)^2 , f(x)=$?
I saw someone solved it like this:
She first takes that $x = u$, then equation looks like this $3f(u) + f(2-u) = (u)^2$
Then she takes that $x = 2-u$, then equation ...

Can some explain how the highlighted implication is so obvious i cant seem to get there from previous step

Motivation:
Consider $\mathcal{X}=(X, +)$, where $X=\{-1, 0, 1\}$ and $+$ is standard addition. Then $\mathcal{X}$ is associative (where defined) but not closed.
NB: There is an identity element in $...

Since 0! = 1 and 1! = 1, am I correct in saying that the factorial function is not injective over the Reals?

$p(x)$ is a polynomial with real coefficients, such that:
$$2p(x^2)=p(x^2+1)+(x^2+1)$$
Thus, $$2p(x)=p(x+1)+x+1$$ and $$2p(x-1)=p(x)+x$$
From that, what I did was write p(x) as $$a_n x^n+a_{n-1}x^{n-...

Let $R:=C(\mathbb R,\mathbb R)$ be the ring of continuous $\mathbb R$-valued functions on $\mathbb R$. Is every prime ideal in $R$ maximal?

Please can you help me to prove that:
$$f^{-1}\left( f \left( f ^{-1} \left(f(A) \right) \right) \right) = f ^{-1}(f(A))$$
where $A \subseteq X$ and $f(A)$ is the image of $A$.

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$.
I already know there is no function that is ...

I am studying limits, continuity and derivability and I have got a continuous function $f\colon \mathbb R \to \mathbb R$ such that $f''(x)= f(x)$. How do I find $f(x)$ from this information? This is ...

What does argument of a function mean? I read Wiki etc and I'm still not sure.
Is Argument and Domain one and same thing?
If Domain corresponds to Range? Do we have a term corresponding to Argument ...

Given that $F_1, F_2, F_3, F_4$ are applications of $F_i: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ which are defined as $F_1: (x, y) = (x, y), F_2: (x, y) = (-x, y), F_3: (x, y) = (x, -y), F_4: (x, y) = ...

Of all the subsets of {1,2,3,4,5,6,7} a non-void subset is randomly selected. The probability that it does not contain two consecutive numbers is equal to

Consider the following integration:
$$
I(x_0|c, d) = \int_{x=-\infty}^{x_0} (cx - d) \mathcal{N}(x; \mu, \sigma^2) \,dx
$$
In this notation, $\mathcal{N}(x; \mu, \sigma^2)$ is a normal distribution: ...

Let $f(x,y)$ be a non-separable, non-negative real-valued function, that is jointly concave in $x$ and $y$. We want to maximize $f(x,y)$ over $x$ and $y$.
Is the sequential maximization $$\max_{x} \...

There was this question asked yesterday here:
Link: Evaluating $\lim\limits_{x→∞}\left(\frac{P(x)}{5(x-1)}\right)^x$
Consider $P(x)= ax^2+bx+c$ where $a,b,c \in \mathbb R$ and $P(2)=9$.
Let $\...

Suppose there exists a continuous function satisfying the functional equation, $ f(g(x))=f((h(x))+ f_1(x)$.
Now the way I like to tackle this kind of problem is assuming $f(x)= \sum a_if(x)$. Where ...

I'm having trouble finding $\lim_{x\to \infty} {x\over \sqrt{1+x^2}}$ by using the formal analysis proof i.e. $\forall \epsilon>0$ $\exists N: \left|f(x)-L\right|<\epsilon$ $\forall x>N$. I ...

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