Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

Filter by
Sorted by
Tagged with
4 votes
1 answer
69 views

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 ...
1 vote
1 answer
36 views

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: $$ V = \{y~|~y''=0~~\text{and}~y \in C^2(- \infty , \infty )\} $$
0 votes
0 answers
14 views

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 ...
0 votes
0 answers
30 views

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?
0 votes
0 answers
38 views

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 votes
0 answers
71 views

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 votes
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&...
0 votes
0 answers
45 views

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 ...
0 votes
0 answers
26 views

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 ...
  • 1
-1 votes
1 answer
18 views

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 votes
0 answers
27 views

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
0 votes
1 answer
46 views

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 votes
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 ...
  • 41
0 votes
0 answers
20 views

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 ...
  • 41
-2 votes
0 answers
27 views

Limit Derivation and Proof (Rigorous ε-δ Proof) [closed]

Could someone, please, provide me with a rigorous ε-δ proof for the following: $$ \lim_{x \rightarrow 4^{-}}\frac{\sqrt{x}-1}{(x^{2}-16)^{5}}=-\infty $$ ?
2 votes
2 answers
82 views

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 views

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 ...
0 votes
0 answers
26 views

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 \...
  • 645
4 votes
0 answers
102 views

$f(m+n)+f(mn)=f(m)f(n)+1$ [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 views

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 views

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 ...
0 votes
0 answers
18 views

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,...
1 vote
2 answers
100 views

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 $x\in\mathbb{R}$ ...
  • 631
0 votes
0 answers
57 views

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 $\...
0 votes
0 answers
24 views

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) = ...
0 votes
1 answer
71 views

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 ...
  • 153
-3 votes
1 answer
58 views

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
583 views

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 views

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 ...
  • 101
2 votes
0 answers
76 views

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 views

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, ...
  • 23
0 votes
0 answers
15 views

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 views

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 ...
0 votes
0 answers
36 views

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 ...
0 votes
0 answers
16 views

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?
  • 101
4 votes
1 answer
75 views

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
5 votes
3 answers
67 views

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 votes
0 answers
44 views

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
27 views

$U$ 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 $...
0 votes
2 answers
31 views

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 ...
-2 votes
0 answers
28 views

surjective or not [closed]

Can anyone provide me with the proof that the following function is surjective
  • 17
0 votes
1 answer
62 views

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....
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: ...
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 views

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 ...
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)...

1
2 3 4 5
647