Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
32,341
questions
4
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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 ...
- 73
1
vote
2
answers
30
views
Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
- 320
1
vote
1
answer
36
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Finding a Basis to a Vector Space of Functions
I'm having some difficulty devising how one finds a linearly independent set of vectors to span this space:
$$ V = \{y~|~y''=0~~\text{and}~y \in C^2(- \infty , \infty )\} $$
- 11
0
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0
answers
14
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Is considering only one output for each input for function a convention, or is it a reflection of the reality of the world?
I'm reading algebra from brilliant and it defines function as:
A function is a relationship where every allowed input results in exactly one output — there's no choice.
My question is, what makes ...
- 119
0
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0
answers
30
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Angle brackets around the function < f(x) > notation meaning
I've seen it several times in papers, but I can't understand the point. I've seen the same equations without the brackets as well. Are they trying to emphasize something?
- 151
0
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0
answers
38
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Finding a function with a functional equation.
A function $f:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ which satisfies the functional equation $$f((c+1)x+f(y))=2cx+f(x+2y)$$
Where $c$ is a positive constant, Find all such possible functions.
I ...
- 130
0
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0
answers
71
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On an exotic type of "mathematical mean"
Background, motivation and the problems
Most of us are familiar with the arithmetic mean (AM) - the simplest and most basic type of mathematical mean. The arithmetic mean can be said to be based on ...
0
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0
answers
48
views
Are all surjections continuous functions?
I am reading Wikipedia page about affine transformations, and have not read it English version, but in Russian, it written, that "continuity follows from the definition in a not quite trivial way&...
- 201
0
votes
0
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45
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Best approach for creating a function from multiple inputs/outputs?
I have a list of inputs and outputs, and I need to somehow find a function that will match them. There's two inputs and one output per set. At first I considered using a Neural Network, but it might ...
- 101
0
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0
answers
26
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Legendre transformation of Geometric Random Variable
I am wondering what the Legendre transform of a Geometric r.v with success parameter $p$ is. I have that the logarithmic moment generating function is $\Lambda(t)=(t+\log p)-\log(1-(1-p)e^t)$ but ...
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-1
votes
1
answer
18
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Property of linear multidimensional function? [closed]
Is it true that if $f(x,y)$ is a linear on both x and y
$f(ax,by) = af(x,by)+bf(ax,y)$?
- 21
0
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0
answers
27
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What do you call the point on the function where the gradient of the graph changes from increasing to constant? [closed]
Essentially, what do you call a point on the graph where the second derivative changes from greater than 0 to exactly 0?
i.e. the graph is initially concave upwards then becomes linear
- 1
0
votes
1
answer
46
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Show that if $X$ is a solution of $X'=A(t)X$ with $A$ a matrix and $X(0)$ with positive coefficients, $X(t)$ has positive positive coefficients
Let $A : \mathbb R^+ \to M_n(\mathbb R^+) \in C(\mathbb R^+,M_n(\mathbb R^+) )$ and $X$ a solution of $X'=AX$ such that $X(0)$ with positive coefficients $(\ge 0)$.
I need to show that : $\forall t\in ...
4
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0
answers
43
views
$g$ attains both maximum and minimum over $\mathbb{R}$. $f$ is continuous on range of $g$. Does $f\circ g$ necessarily attain maximum on $\mathbb{R}$?
Let $g:\mathbb{R}\to\mathbb{R}$ be a function (not necessarily continuous) which attains both its maximum and minimum on $\mathbb{R}$. Let $f:\mathbb{R} \to \mathbb{R}$ is a function which is ...
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0
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0
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20
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Can a continuous function f on a compact, convex set D have a discrete range with more than 2 elements?
Give an example of a compact set $D \subset \mathbb{R}^n$ and a continuous function $f:D \to \mathbb{R}$ such that $f(D)$ consists of precisely $k\geq 2$ points. Is this possible if D is convex? Why ...
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0
answers
27
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Limit Derivation and Proof (Rigorous ε-δ Proof) [closed]
Could someone, please, provide me with a rigorous ε-δ proof for the following:
$$
\lim_{x \rightarrow 4^{-}}\frac{\sqrt{x}-1}{(x^{2}-16)^{5}}=-\infty
$$
?
- 1
2
votes
2
answers
82
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Understanding a proof that, if $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$, then the minimum value of $|a-b|$, over distinct reals $a$ and $b$, is $2$.
I saw this epic question in Advanced Problems in Mathematics by Vikas Gupta:
If $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then find the minimum value of $|a-b|$, where $a$ and $b$ are distinct real numbers....
1
vote
1
answer
26
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About the theorem that links limits of sequences and limits of functions, and the arbitrary delta.
First things first, I apologize if at any point I spell any formulas incorrectly or if at any point I name something wrongly (I am taking the class in another language and I am translating everything ...
- 25
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0
answers
26
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Show Gateaux differentiability
The partially linear model, $\theta_0$ and $g_0,m_0 \in F\subseteq \{f|f:\mathbb R^d \rightarrow \mathbb R\}$ where $F$ is a fixed function class, and we consider a random vector $Z=(D,X,Y) \in \...
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4
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0
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102
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$f(m+n)+f(mn)=f(m)f(n)+1$ [closed]
How to find all functions $f:R\rightarrow R$ such that
$f(m+n)+f(mn)=f(m)f(n)+1$
I I've tried a lot, but I didn't find a solution
I only know that $f(x)=1 $ and $f(n)=n+1$ are solutions
4
votes
1
answer
69
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non-decreasing surjective map from reals to rationals [closed]
I have a feeling that no such map exists, i.e., there is no non-decreasing surjective function from $\mathbb{R}$ to $\mathbb{Q}$. But I am just unable to write an argument. Any hint or sketch of proof ...
- 2,437
3
votes
2
answers
57
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Does a bijective function exists behind every recurrence relation?
Consider these 2 questions where recurrence relations can be applied:
Q1) Given an (nxm) where n denotes rows and m denotes columns of a grid, find the number of unique paths ($a_{n,m}$) that goes ...
- 53
0
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0
answers
18
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Show that $\langle y^1 - y^2 , x^1 - x^2 \rangle \geq 0 $
I am trying do this exercise:
Exercise: Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ a convex funciton. Show that:
$$\langle y^1 - y^2 , x^1 - x^2 \rangle \geq 0, \qquad \forall x^i \in \mathbb{R}^n,...
- 35
1
vote
2
answers
100
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Proving that if $f(x)=f(e^tx),$ then, $f$ is a constant function
This question was asked in an entrance examination for ISI (Indian Statistical Institute):
Q. If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function such that $f(x)=f(e^tx)$ for all $x\in\mathbb{R}$ ...
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0
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57
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If s and t are positive step functions defined on an interval [a,b], is s/t also a step function?
I think it is, assuming that both functions are defined on the same partition (joint refinement), if we let s = $\sum_{i=1}^n c_i1_{[p_{i-1},p_i]}$ and t =$\sum_{i=1}^n d_i1_{[p_{i-1},p_i]}$ then $\...
- 35
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0
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24
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Representing matrix of $id_V , f$ and $f^2$
Hey I have this exercise and I want to see if what I have done is correct.
Let $v_1$ and $v_2$ be a basis of the vector space $V$. Let $f: V \rightarrow V$ be the $K$-linear function with $f(v_1) = ...
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1
answer
71
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Prove: $\mathbb{C}^{*} \cong U_1 \times \mathbb{R}^{*}_{> 0}$ (isomorph to circle group)
Define the circle group $U_1 = \{z \in \mathbb{C} \mid |z| =1 \} \subset \mathbb{C}^*$. Prove: $\mathbb{C}^* \cong U_1 \times \mathbb{R}_{>0}^*$.
I would like to know if I did it correct. I ...
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votes
1
answer
58
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Measure body surface area of a human body [closed]
**Edit: Sorry, i was unaware that such questions are frowned upon and go against community guidelines and i didn't mean to come off that ignorant. Mods can delete thread.
so i've been given this ...
4
votes
3
answers
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Find the derivative of a difficult integral
This exercise is very difficult for me.
Find the derivative of the function:
$$
\int_{0}^{\ln x}f(t) dt
$$
I use this formula:
$$
\int(b(x)) \cdot b'(x) - \int(a(x)) \cdot a'(x)
$$
where this is $b'(x)...
- 49
0
votes
1
answer
38
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Can I rewrite an exponential with product of powers as product of two terms?
I have what I thought was a relatively simple problem but cannot quite pinpoint the answer. I want to do a regression of something like $a \times \mathrm{e}^{-b/x}$. However, I can only regress the ...
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2
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0
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Name for symmetric functions, $f(x, y)=g(x) h(y)+h(x) g(y)$.
I was playing around with some ideas and thinking that this function
$$f(x, y)=x \sqrt y+y \sqrt x $$
looked like it might be fun to explore.
But then I considered looking at the set of functions with ...
- 488
1
vote
1
answer
46
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What is the codomain of $g(x) = x^2 + 1$?
I do not understand the first example right after the topic of function composition is introduced in Mathematik für Ingenieure by Thomas Reissinger.
There is a function $g : \mathbb{R} \rightarrow [0, ...
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0
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15
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I wrote this code for a natural cubic spline and the result at 5 is not right what is the error in my code?
import numpy as np
def The_CubicSpline(t,bt):
L= len(t)
h=np.zeros(L-1)
for i in range(L-1):
h[i]=t[i+1]-t[i]
...
- 1
1
vote
1
answer
38
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Need help understanding step functions notations and concepts
I am confused on the notations of step functions.
If we have a sequence of step functions $\{s_n : [0,1] \rightarrow \mathbb R \}_{n = 1}^{\infty}$ , what would $s_n(0)$ look like, as in if I plug ...
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0
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36
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Rational fractions as geometric predicates
I read article about restricted voronoi diagram. In chapter 3.4 Exact predicates I saw the following: "sideA, sideB, sideC are rational fractions of low-degree (resp. 2, 4/2 and 6/4)." sideA ...
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0
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Expression that encapsulates both break points and end points on a piecewise linear function [closed]
Does an expression exist that encapsulates both break points and end points when describing a piecewise linear function valid only on a given interval?
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4
votes
1
answer
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How to Check monotonicity of a function
Given a function, $f$ be defined on $[0,1]$ by
$$f(x) = \begin{cases} 0 & x=0 \\ \frac1{2^{n-1}} & \frac1{2^n}\lt x\le\frac1{2^{n-1}}, n\in\Bbb N\end{cases}$$
If I take $n =1$, then
$f(x) = 1$ ...
- 43
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votes
3
answers
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Given a symmetrical function, how to solve so symmetrical point lands on $x$? [closed]
Say I have the following curve:
$$y = k_1 + \dfrac{k_2}{1-x} \left\{0\leq x\leq1\right\}$$
If I say that:
$k_1=0.038$
$k_2=0.002$
I get this:
How do I?
a) Find the point of symmetry (somewhere in ...
-1
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0
answers
44
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What can you say about the value of a polynomial at x = 1 when some conditions about the polynomial are given?
A polynomial P(x) with real coefficients has the property that P''(x) ≠ 0 for all x. Suppose P(0) = 1 and P'(0) = –1. What can you say about P(1) ?
(A) P(1) ≥ 0
(B) P(1) ≤ 0
(C) P(1) ≠ 0
(D) –1/2 < ...
1
vote
1
answer
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$U$ be the set of all sequences of real numbers $(x_n)_{n\in \mathbb{N} }$ with $x_n + x_{n+1} = x_{n+2}$ find $\dim(U)$ and find a basis
Let $F = Abb(\mathbb{N}, \mathbb{R})$ be the $\mathbb{R}$-vector space of sequences of real numbers.
Let $U$ be the set of all sequences of real numbers $(x_n)_{n\in \mathbb{N} }$ with the property
$...
- 388
0
votes
2
answers
31
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Can a function that maps one-one have more than one point that lies on the line of symmetry $y=x$?
As far as I'm aware, a function $f$ which maps one-one must have an inverse function $f^{-1}$, and the graphs of these functions must intersect at some point (if their domains permit this) on the line ...
- 11
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0
answers
28
views
surjective or not [closed]
Can anyone provide me with the proof that the following function is surjective
- 17
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votes
1
answer
62
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How to find domain of a function?
I came across a question, where in I was asked to find out the domain of the function
$$f(x)= \sqrt\frac{(1-|x|)}{( 2-|x|)}\;$$
Now I can actually put any values and get an answer out of that function....
- 21
0
votes
0
answers
18
views
Precedence Function Call vs Power
I am currently programming a terminal calculator, and have come across an interesting problem with the precedence of function calls. How should this be evaluated:
...
- 1
0
votes
2
answers
84
views
How are the domain and range of a quadratic function determined?
How are the domain and range of a quadratic function determined?For example, what are the domain and range of $x^2-4x+10$
- 19
0
votes
2
answers
82
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Find $f^{(25)}(x)$ if $f(x)=x^{-3}$ by first finding general solution.
I don't get it what is the question asking for either composition or derivative of f 25th times.
Solution 1.
Sol:
$f(x)=x^{-3}=1/x^3$
$f^{(2)}(x)=f(f(x))=f(1/x^3)=1/(1/x^3)^3=1/(1/x^9)=x^9$
$f^{(3)}(x)...
2
votes
2
answers
71
views
When does the equations have $1,2,3$ solutions?
There is given an equation,
$$ \frac{x^2-x+1}{x^2+x+1}=kx+1$$
When does this solution have one, two, three real solution(s) in $x$.
My Approach:
The above equation can be rewritten as,
$$\frac{-2x}{x^...
- 130
1
vote
0
answers
47
views
What is the type of discontinuity of $e^{\frac{1}{x}}$ at zero?
The limits of this functions at zero are:
$\lim_{x \to 0^+} e^{\frac{1}{x}} = \infty $ an infinity discontinuity
$\lim_{x \to 0^-} e^{\frac{1}{x}} = 0 $ a removable discontinuity
The question is: Is ...
- 185
1
vote
0
answers
17
views
Prove that for L all local minima are global.
For a vector v consider the rank-$1$ PCA loss function,
$$L(w) = ||{vv^T - ww^T}||_2^2$$
Prove that for L all local minima are global.
To prove that all local minima of the function $L(w)$ are global,...
3
votes
1
answer
33
views
Integration a zero valued function product other function
Suppose I have a function $f(x) = 0$ for all $x$ except $0$. Now, $f(x) g(x)$ will also be zero for all $x$ except $0$ (or may be at $0$). Now, if try to find the value of $\int_{-\infty}^{\infty} f(x)...
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