Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
32,363
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Rudin PMA 4.20 - how can this function be unbounded ? Considering Rudin hasn't introduced "divergence" of functions yet in the chapter.
Here is the very beggining of Rudin's Principles of Mathematical Analysis 4.20 theorem:
Let $E$ be a noncompact and bounded set in $\mathbb{R}^1$. Then there exists a continuous function on $E$ ...
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1
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When raising a bracket (of a function like $\ln$) to a power, is the power applied before the ln operation?
I've seen sources that apply the $\ln$ function before the power in $\ln(x-1)^2$ for example and others where it is applied after the power. Which is correct?
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Composition of functions where f(g(x^2)), how do you handle the g(x^2) function?
Plugging the question into symbolab, it only applies the square to the x within the g(x) function. Example: f(x) = x^2 - 2, g(x) = x - 7. The g(x^2) = x^2-7. f(g(x^2)) becomes x^4-14x^2+47. The ...
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0
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Mathematica plot function of this
Descriptive function of motion of a rigid rod around an axis and, by analogy, around a cylinder and the volume "swept" from it
I kindly ask for your help for the "rust" that, after ...
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2
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29
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$f(U_{\alpha \in I} X_\alpha) = U_{\alpha \in I} f(X_\alpha)$
So I figured this:
Take an arbitrary x s.t. $x \in U_{\alpha \in I} X_\alpha$.
So f(x) $ \in f(U_{\alpha \in I} X_\alpha)$ which gives us f(x) is in at least one $f(X_\alpha)$ therefore, $f(x) \in U_{\...
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1
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Descriptive function of motion of a rigid rod around an axis and, by analogy, around a cylinder and the volume "swept" from it
I kindly ask for your help for the "rust" that, after many years of mathematical inactivity, I have regarding the schematization of physical problems.
Problem, as from the title, concerns ...
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0
answers
14
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What can I do to the function of an archimedian spiral to make the sprial eccentric?
I know the function of an archimedian spiral is r=theta. Is there a way to "squish" the spiral to make it thinner or taller? I'm trying to code something ...
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2
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Showing that $\sqrt[3]{x + \sqrt{\frac{x^3}{6^3} + x^2}} + \sqrt[3]{x - \sqrt{\frac{x^3}{6^3} + x^2}}$ is bounded by 4
I'm trying to show that 4 is an upper bound for $f(x)=\sqrt[3]{x + \sqrt{\frac{x^3}{6^3} + x^2}} + \sqrt[3]{x - \sqrt{\frac{x^3}{6^3} + x^2}}$. This can easily be seen using tools like WolframAlpha. ...
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1
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A combinatorics problem dealing with injective functions
I'm trying to solve this combinatorics problem:
If $A=\{1,2,3,4\}$, $B=\{1,2,3,4,5,6\}$ and $f:A\to B$ is an injective mapping satisfying $f(i)≠i$ for all i∈A, then the number of such mappings are:
...
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What do you call two surjective functions that have the same codomain?
Let $f$ be a surjective function from $D_f$ to $C$, and let $g$ be a surjective function from $D_g$ to $C$ such that $C = \{ f(d) | d \in D_f \} = \{ g(d) | d \in D_g \}$. Assume $f \neq g$. Is there ...
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Function defined in all but 3-th quadrant [closed]
Could you please give example of a function, wich is defined in all 1, 2 and 4 quadrants, but not in 3?
I.e. have no negative real roots nor negative real values.
Strictly saying:
$$
dom f =
\begin{...
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Looking for a function that its graph looks like this. [closed]
THIS IS THE LINK FOR THE GRAPH
I'm looking for any function that its graph may look like the image above. It would be amazing if the function can be modified to move around in term of the X axis.
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1
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Polynomial functional equations
Find all polynomials $f:R→R$ such that $f(x+2)=f(x)+2$
Find all polynomials $g:R→R$ such that $g(2x)=2g(x)$
Since the functions must be polynomials, I tried using an argument by degrees, but this did ...
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0
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39
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Prove the sequences are even aside from 1st term
Consider the following two alternating sequences: $$ A=\bigg\lbrace f^{(1)1}(x),\cdot\cdot\cdot \bigg \rbrace \bigg|_{x=1/e}=\bigg \lbrace-1,2,-6,32,-320,4452,-70798, \cdot\cdot\cdot \bigg \rbrace$$
$$...
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1
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Is the definition $f(x+1) = x$ a valid function definition? If not, why not?
I encountered the following informal brainteaser:
Is the definition $f(x+1) = x$ a valid function definition?
If not, why not?
Not having much formal training in mathematics,
I figured this was ...
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1
answer
42
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Lambda calculus novice seeking help with defining isempty for list representation
I'm exploring the concept of untyped lambda calculus and I'm facing a challenge with defining the isempty function. I have a few definitions that I'm working with, which are:
...
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2
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What are the procedures to analyze the similarity of graphs?
This question is taken from a very challenging calculus problems book called the Advanced Problems in Mathematics by Vikas Gupta
And I have absolutely no clue on how to approach questions of the kind....
3
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2
answers
47
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Logical approach of a simple combinatorics problem [duplicate]
Consider all functions f:{1,2,3,4}→{1,2,3,4} which are one-one, onto
and satisfy the following property: if f(k)is odd then f(k+1)is even,
k =1,2,3. Then, number of such functions
So this is how I ...
6
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1
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157
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Show that the function is strictly increasing.
How should I show that the function $f$ defined below is strictly increasing for $x\in(0,1)$?
I have considered its first derivative, but it seems too complicated to deduce $f'>0$ from there.
$f(x)=...
0
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0
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Bound of a ratio of summations and products of terms of bounded ratios
Given $D_1, D_2 \in \mathcal{S}$ and given a space $\mathcal{M}$ such that $\mathcal{S} \subset \mathcal{M}$, I have the following function:
$$
\frac{\sum_{\mathcal{S}_{i} \in \mathcal{M}} \left[ f(...
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2
answers
58
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How do I start off with integral functional questions like these? [duplicate]
I saw this question in Advanced Problems in Mathematics by Vikas Gupta
If $f^{\prime}(x)=f(x)+\int_0^1 f(x) d x$ and given $f(0)=1$, then
$\int f(x) d x$ is equal to :
I have no clue to on how to ...
4
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1
answer
71
views
How to find an explicit formula for this function?
Let us take
$$
\mathbb{N} := \{ 1, 2, 3, \ldots \},
$$
and let the function $f \colon \mathbb{N} \longrightarrow \mathbb{N} \times \mathbb{N}$ have the following values:
$$
\begin{align}
& f(1) :=...
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Are there any examples of non-piecewise, non-analytic, smooth functions with converging Taylor Series?
There is—for example—the piecewise function where $f(x) = e^{(-1/( x^2 ))}$ if $x \neq 0$ and $f(x) = 0$ if $x = 0$, where the Taylor Series (centered at $x = 0$) becomes $0 + 0x + 0x^2 + 0x^n +…$, ...
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Prove that $|A| = |B|$ [closed]
$A$ is the set containing all bijections from $\mathbb{N}$ to $\mathbb{N}$. $B$ is the set containing all total orders $\preceq$ such that $(\mathbb{N}, \preceq)$ is well-ordered. How do I prove that $...
2
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0
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Are the Dominating Families Of Functions (Domsets) Uncountable?
Preface :
For $f, g: \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$, we write $f \leqslant^* g$ for
$$
\exists m \in \mathbb{Z}_{\geqslant 0} \forall n \in \mathbb{Z}_{\geqslant m} f(n)...
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3
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Smooth monotonic function $f: [-1, 1] \to [-1, 1]$
I'm looking for a function $f_t: [-1, 1] \rightarrow [-1, 1]$ parameterized by a "threshold" $t\in(-1, 1)$ that meets the following constraints:
$f(x)$ is smooth and monotonically ...
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Antisymmetric polynomials in two variables
In Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials it is argued that a symmetric polynomial of two variables $x,y$ can be written as a sum over ...
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1
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Find the maximum value of $|b|+|c|$.
Let $\mathit{f}:\mathbb{R}\to\mathbb{R}$, $\mathit{f}(x)=ax^2+bx+c$ such that $|\mathit{f}(x)|\le1$ for any $x\in\mathbb{R}$ with $|x|\le1$. Find the maximum value of $|b|+|c|$. I presume that if $x=0$...
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Difference in usage between function, mapping, functional, form, and operator?
The word function has many synonyms (or close to synonyms), including:
map
functional
form
operator
transformation
What is the difference, in meaning or usage, between them? I understand that exact ...
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Suppose that $A$ and $B$ are infinite sets such that $|A| = |B|$, prove that the set of all bijections from $A$ to $B$ is uncountable [duplicate]
According to Cantor's definition of same-cardinality, the set of all bijections from $A$ to $B$ must be non-empty. And we also know that for finite sets $A$ and $B$, the set of all bijections from $A$ ...
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How to find $p(n+2)$ for a $n$th degree polynomial?
//This question was solved as by the hints given in the comments section, thank you to all!
Here's the exact question:
If $p(x)$ denotes a polynomial of degree $n$, such that $p(k) = 1/k$ for $k = 1, ...
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3
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Polynomial functional equation
Find all polynomials $f : R \rightarrow R$ such that $f\left(\frac{1}{x+1}\right)=\frac{1}{f(x)-1}$.
Since the functions must be polynomials, I tried using an argument by degrees, but this did not ...
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1
answer
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A formula for this sum : [closed]
I need a simplfied formula for this sum :
$$\sum_{i=0}^{n}{(a + iΔ)^y}$$
Thanks.
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1
answer
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Is ∞ a limit point of $\mathbb R$ ? If not, how to understand Rudin's definition at the beggining of chapter $4$ (PMA)?
I am starting reading the $4^{th}$ chapter of PMA from Walter Rudin.
The chapter is about continuity and it defines $$\lim_{x\to a} f(x)$$ (for a function mapping a metric space $E$ into a metric ...
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How to solve root function? [closed]
How to solve x? I cant get right answer
$$1-4(1-x)^{1/2} = 4x$$
Right answer is 3/4
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how can apply the circle criterion?
My question is:
If I have a transfer function $(G(s)=\frac{2}{(s+1)^2})$ like the one in picture , and I want to determinate the lower value on the real axis of the function in the nyquist diagram, in ...
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Is there a way to make an if, then statement in an equation? [closed]
I basically need the equation to be $d=f(96+(8⋅f/2))$ where I need it to be, if $f$ is odd then, subtract $4$ from $d$.
I think the best answer should be how to turn the sentence into the equation, ...
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2
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Proof that $a^n$ is well-defined
Consider a non-empty set $X$ and an associative operation $*$ on $X$ with identity element $e$. For $a\in X$ define
$$a^0:=e,\quad a^{n+1}:=a^n*a,\quad n\in\mathbb{N}.$$
Then $a^n$ is well-defined ...
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Can surjective functions be strictly monotonous? [closed]
I know that it is a property of injective functions that they must me strictly monotonous, but can surjective functions be strictly monotonous as well?
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Two subsets $A$ and $ B$ of the $(x,y)$ plane are said to be $equivalent$ if there exists a function $f:A\to B$ which is both one-to-one and onto.
Two subsets $A$ and $ B$ of the $(x,y)$ plane are said to be equivalent if there exists a function $f:A\to B$ which is both one-to-one and onto.
$(i)$ Show that two line segments in the plane are ...
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uniform converge
Question: Considering the uniform convergence of series function $\sum_{n=1}^{\infty}\dfrac{e^{inx}}
{n^{\alpha}}$, with $\alpha\in\mathbb{R}$, $x\in[a,b]\subset (0,2\pi)$.
Series function converges ...
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2
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$g = -e^{-f(x,y)}$ is concave if we know that $f(x,y) = 2x+y-x^2$ is concave.
Let $f(x,y) = 2x+y-x^2$ be a two variable function.
(a) Show that $f$ is concave.
(b) Show that $g = -e^{-f(x,y)}$ is concave.
I have proved part (a) using Hessian matrix.
We have $f_x = 2-2x$ and $...
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Terence Tao's definition of function equality
In his Analysis I book, Terence Tao defines two functions $f,g:X\to Y$ to be equal if $f(x)=g(x)$, for all $x\in X$. After giving some examples of this concept, he then says:
This notion of equality ...
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Interpolating functions with the property $f(x) = x^{g(x)} f(x-1)$
There is a video on YouTube detailing how one can extend the factorials to the reals and arrive at the function
$x! = \displaystyle \lim_{n \to \infty} n^x \prod_{k=1}^{n} \frac{k}{x+k}$
Following the ...
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0
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Universal approximation related to activation regions
We assume we have an interval $I=[a,b]$. We define $C(I)$ to be the set of continuous functions on $I$. We further define the set of one-hidden layer neural networks
$$NN(H,\theta)=\left\{ f_{\theta}=\...
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1
answer
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periodic function is constant
Let $g:\mathbb{R} \rightarrow \mathbb{R} $ be a function with intermediate value property, such that $\lim_{x\to\infty} g(x) = \infty$. Prove that if $f:\mathbb{R} \rightarrow \mathbb{R} $ is a ...
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Let $a, b, c$ be real numbers such that $3b > a^2$. Then the function g : $\Bbb R \to \Bbb R$ given by $g(x) = x^3 + ax^2 + bx + c$ is
Let $a, b, c$ be real numbers such that $3b > a^2$. Then the function g : $\Bbb R \to \Bbb R$ given by $g(x) = x^3 + ax^2 + bx + c$ is
(A) one-one and onto
(B) onto but not one-one
(C) one-one but ...
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1
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0
answers
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Prove that the square root is a function [closed]
So I know that square root is defined as:
Say $y=x^2$ then $\sqrt{y}=\sqrt{x^2} = |x|$
With this definition how do you prove this defines a function? (Although, I guess I wouldn’t know how to prove ...
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2
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40
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Level set of functions $f(x,y) = x+y$ and $g(x,y) = xy$ with $M := \{ (x,y) \in \mathbb{R^2}: x^2+y^2 \leq 9 \}$
Given are the functions
$$f(x,y) = x+y \\ g(x,y) = xy$$
and the set $$M := \{ (x,y) \in \mathbb{R^2}: x^2+y^2 \leq 9 \}$$
The question was to visualize the level set and to mark the points
$$\max_{(x,...
4
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1
answer
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What does the equation $x^2+y^2=r^2$ represent when $x, y, r$ are complex numbers?
I know this question is vague or maybe broad and subjective.
But, I am interested in studying the equation $x^2+y^2=r^2$ when $x,y,r$ are complex numbers.
What are a few directions that I can follow ...
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