Questions tagged [functions]

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

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Proof that $f(x,t)$ can be written as $g(x-at)$ given certain condition.

Let's say I have a function $f(x,t):R^2\to R$ and $a = - \frac {{\frac{\partial f}{\partial t}}} {\frac{\partial f}{\partial x}}$ is true for every element of D, where $a$ is a real constant. How can ...
hector's user avatar
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Notation for map evaluated at all elements of a set?

Consider a map $f : X \to Y$ and some given set $A \subseteq X$. I would like to introduce the notation $f(A) = \{ f(a) \vert a \in A \} \subseteq Y$, i.e. the set of output given the elements of $A$ ...
Bart Wolleswinkel's user avatar
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1 answer
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how to prove a function exists (simple example)

This question arises because I am a self-teaching beginner - and am new to proof patterns and strategies. I asked a similar question but the replies were overly complex because the exercise was ...
Penelope's user avatar
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Does a Schwartz function, part of which is sine carve exist? [closed]

I want to find Schwartz function, part of which is sine carve in a certain interval. If there are multiple function satisfying this condition, please answer simple function as much as possible.
W1234qq's user avatar
1 vote
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A sequence of non-trivial continuous functions $f_{n+1}(f_{n+1}(x))=f_{n}(x)$, $f_2(x)= \frac{1-x}{1+x}$.

Just for curiosity I was trying to find a sequence of non-trivial continuous functions at $\mathbb{R}$ except finite many points such that $f_{n+1}(f_{n+1}(x))=f_{n}(x)$, $f_1(x)=x$ and by non trivial ...
pie's user avatar
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Isolating $z$ in the equation $x - 1 = - \frac{1-y^{z+1}-0.5(1-y^z)}{(1-y)y^z}$

I have a formula with multiple unknowns: $$x - 1 = - \frac{1-y^{z+1}-0.5(1-y^z)}{(1-y)y^z}$$ The way it is setup now allows to easily calculate $x$, but I would like to reformulate it to isolate $z$, ...
Apo's user avatar
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How to show there is (no) tie of certain functional form $\frac{x_{1}e^{\mu_{1}}-x_{2}e^{\mu_{2}}}{e^{\mu_{1}}-e^{\mu_{2}}}$.

I'm trying to prove or disprove the following. For any $\mu_1<\mu_2<\mu_3$ and $x_1<x_2<x_3$, there is no three way ties of $$\frac{x_{1}e^{\mu_{1}}-x_{2}e^{\mu_{2}}}{e^{\mu_{1}}-e^{\mu_{2}...
user23329063's user avatar
2 votes
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If $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is defined by $f((a,b))=b$, then does $f \circ f$ exist?

Let $f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $f((a,b))=b$. I am confused as to whether $f \circ f$ exists here. I was taught that for this composite function to exist, ...
scob_'s user avatar
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how to prove a function exists? (Tao Analysis I 3.5.7)

Question: How does one prove a function exists? I am a self-teaching beginner, new to proof methods (eg contradiction, contraposition, induction, etc). I am currently doing Tao's Analysis I ex 3.5.7 ...
Penelope's user avatar
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What are the roles of $x$ and $y$ in the functional equation: $f(x+y)=f(x)+f(y)$ for all $x,y$ in R?

I am slightly confused by the roles of input variables for certain functional equations. Kinda new to this study of functions. Knowing the role of variables in standard equations such as $ax+by-c=0$, ...
webtolight's user avatar
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Proving $\int_1^x \lfloor u \rfloor(\lfloor u\rfloor+1)f(u)\,du=2\sum_{i=1}^{\lfloor x\rfloor}i\int_i^xf(u)\,du$ for continuous $f$, with $x>1$

Let $f(u)$ be a continuous function; and, for any real number $u$, let $\lfloor u\rfloor$ denote the greatest integer less than or equal to $u$ (aka, the floor of $u$). Show that for any $x>1$, $$\...
MathStackexchangeIsNotSoBad's user avatar
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Find the minimum value with inegral [duplicate]

$f(x)$ is a continuous differentiable function on $[0,1]$ such that $f(0)=\frac{-1}{3}$ and $\int\limits_0^1 f(x) dx = 0$. Find $$min \int\limits_0^1(f'(x))^2 dx$$ I need a help
Moahmed Amine's user avatar
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A functional equation: Find all bounded functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(m+n)+f(m−n) = 2f(m)f(n)$, for all integers $m, n$.

The question goes as follows: Find all bounded functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(m+n)+f(m−n) = 2f(m)f(n)$, for all integers $m, n$. Here is how I started: Putting $m=n=0$ gives $2f(...
xoxo's user avatar
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2 votes
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Entire functions which satisfy a coefficient property

Let $f$ be an entire function. Then we can write $$f(z) = \displaystyle \sum_{n=0}^{\infty}c_n z^n$$ for some $c_0, c_1, \ldots$. For a positive real $r$, let $$M_r = \sup \{ |f(z)| : |z| = r \}$$. ...
idk31909310's user avatar
7 votes
2 answers
337 views

Number of One-One Functions

This question has been asked in my exam and I have stuck. The question says:Let $S=\{1,2,3,4,5,6\}$. The number of one-one functions $f$ defined from $S$ to $P(S)$, where $P(S)$ stands for power set ...
20DPCO190 Amanul Haque's user avatar
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2 answers
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Finding an increasing function on $\mathbb{R}^+$ such that $f(0)=0$, $\lim_{x\to\infty}f(x)=1$, and $f\circ\left[(f')^{-1}\right]=\frac1{1+x}$ [closed]

This problem about a composite function is driving me crazy! Can you find an increasing function $f(x)$ on $\mathbb{R}^{+}$ with these properties? $$f(0)=0$$ $$\lim_{x\to+\infty}f\left(x\right)=1$$ $$...
bbecon's user avatar
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Does limit exists or not?

I've been trying to solve with trajectories, but I haven't been able to find an answer. $\lim_{(x,y)\to (0,0)} \frac{x^2y^4}{2x^2-2xy+2y^4}$
Ever David Osorio's user avatar
1 vote
1 answer
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Are $\sin{x}$ degrees and $\sin{x}$ radians considered different functions?

As I understand, functions map elements of a set to another, so they should be indifferent to such a thing as units. If $\sin{\frac{\pi}{2}}$ is defined to be 1 then it should always equal 1, no ...
cabutchei's user avatar
0 votes
3 answers
70 views

Period of function $f(x) = \{x\} + \{x + \frac{1}{3}\} + \{x + \frac{2}{3}\}$

Consider the function $f(x) = \{x\} + \{x + \frac{1}{3}\} + \{x + \frac{2}{3}\}$ Let's break this into three individual functions $g(x) = \{x\}$, $h(x) = \{x + \frac{1}{3}\}$, and $i(x) = \{x + \frac{...
Haider's user avatar
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unsure of my solution to Tao Analysis I 4th ed 3.5.2 (axiomatic set construction, cartesian products, power sets)

I am a self-teaching beginner and unfamiliar with proofs - in particular I find proposals that are intuitively true harder to prove as my brain assumes too much or skips steps. I'd like help with my ...
Penelope's user avatar
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-5 votes
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Functions problem in discrete mathematics [closed]

Give an explicit formula for a function from the set of positive integers to the set of positive integers that is One-to-one, but not onto. Onto, but not one-to-one One-to-one and onto Neither One-to-...
M Wahab's user avatar
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Modeling scoring functions

I'm looking for general direction on topics to explore for this problem. I think I'm not searching for the right statistical concepts and therefore coming up empty handed. Also, I'm a computer ...
Jmoney38's user avatar
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Baby Rudin Theorem $4.14$ [closed]

In Baby Rudin theorem $4.14$, he says: $f(f^{-1}(E))\subset E \quad\forall E \subset Y$ and then $f^{-1}(f(E)) \supset E \quad$ if $\quad E \subset X$. I thought functions were invertible $\iff$ ...
Rudinable's user avatar
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1 answer
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Show that the jump $j_f(c)$ of increasing $f$ at $c$ is given by $\inf\{f(y)-f (x): x < c < y, x, y \in I\}$.

Let $I\subseteq \mathbb{R}$ be an interval and let $f: I \to \mathbb{R}$ be increasing on $I$. If $c$ is not an endpoint of $I$, show that the jump $j_f(c)$ of $f$ at $c$ is given by $\inf\{f(y)-f (x):...
user13's user avatar
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4 votes
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Show that $\frac{\text{quadratic}}{\text{quadratic}}$ with no common factors is many-to-one

Let $${f(x)=\frac{ax^2+bx+c}{dx^2+ex+f}}$$ hence, $${f'(x)= \frac{(2ax+b)(dx^2+ex+f)-(2dx+e)(ax^2+bx+c)}{(dx^2+ex+f)^2}}.$$ If $f$ is not a continuously decreasing or increasing function then it is ...
Daksh's user avatar
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1 answer
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Conditions for $f(x) f(1-x)$ to be concave

Let $f : [0,1] \to [0,1]$. Function $f$ is decreasing and satisfies $f(0) = 1$. Does anyone know any conditions on $f$ that ensure that the function $$g(x) = f(x) f(1-x)$$ is concave on the interval $[...
Matteo's user avatar
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-2 votes
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How do I find $a$ and $b$? [closed]

Find $a$ and $b$, so that $f(x)=\begin{cases}(a+1)e^{x-1}, &x\le1;\\ \sin(x-1)+b\cos(2x-2), &x>1\end{cases}$ is differentiable. I calculated left hand limit and right hand limit, and got ...
allee011's user avatar
1 vote
1 answer
22 views

Existence of a function with certain properties, satisfying a functional equation

I came across the following problem. Let $f(x)$ be a real-valued function defined on the real numbers. If $f(x)$ has five distinct roots and $f(5-x)=f(5+x)$, find the sum of the roots. The problem ...
Cornman's user avatar
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On the definition of 'dependent functions' in the set-theoretic semantics of dependent product types

I'm working in a framework (see e.g. this) where a context $\Gamma$ is interpreted as a set $[\![{\Gamma}]\!]$, a type $A$ in context $\Gamma$ is interpreted as a family of $[\![{\Gamma}]\!]$-indexed ...
user837242's user avatar
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0 answers
42 views

Given the functions $g(x) = \frac{2x - 1}{3x – 4},$ find $g^{-1}(x)$. [closed]

Given the functions $g(x) = \frac{2x - 1}{3x – 4},$ find $g^{-1}(x)$. I don't know how to work it out, so I need help to do so.
Rachael Absolam- Browne's user avatar
-2 votes
2 answers
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When solving for x and y in functions. For example $y= x^2$. Is $x = y^{1/2}$ he same function? Are they equivalent? [closed]

Are $y=x^2$ and $x = y^{\frac{1}2}$ the same function/ are the equivalent. As they each would give the $x$ and $y$ value?
Jacob Taylor's user avatar
2 votes
0 answers
76 views

Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective

Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective. As we know $f(x)$ is surjective, the only task left to prove it bijective is to prove that $f(x)$ is strictly monotonic (...
Darshit Sharma's user avatar
0 votes
1 answer
28 views

Fastest root-finding algorithm for specific sine function [closed]

I'm trying to find an algorithm which will tell me any root of the function sin(πx)² + sin(π(323/x))². With this function I can very confidently isolate an interval in which one of these roots would ...
Canbach's user avatar
3 votes
1 answer
28 views

Which is the smallest integer $c>0$ making $a\cdot (b+c)$ a pairing function for positive integer $a,b$ with $a \le a_{max}$ and $b \le b_{max}$?

The goal is to generate $a_{max} \cdot b_{max}$ different numbers with $$a\cdot (b+c)$$ using an integer offset $c>0$ as small as possible. We know $a \in [1,a_{max}]$, and $b \in[1,b_{max}]$ I'm ...
J. Doe's user avatar
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1 vote
0 answers
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Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.

I got stuck on this question: Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$. I think one direction is ...
Beerus's user avatar
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1 vote
1 answer
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Number theory, induction,algebra

Define $$f(k)=\underbrace{\sqrt{m+\sqrt{m+\sqrt{m+\cdots}}}}_{\text{$k$ $m$'s}}-\underbrace{\sqrt{m-\sqrt{m-\sqrt{m-\cdots}}}}_{\text{$k$ $m$'s}}$$ Given $m$ and $k$ are integers such that $m\ge1$ and ...
Sankalp Kumar Jha's user avatar
1 vote
1 answer
19 views

Smooth functions with vanished derivatives at the boundary

I need to find a function $f$ in $[a,b]$ which satisfy that $$ f(a) = f_1, \\ f(b) = f_2, \\ f^{(j)}(a) = 0, \quad j = 1,...,k, \\ f^{(j)}(b) = 0, \quad j = 1,...,k, $$ where $f^{(j)}$ is the j-th ...
luyipao's user avatar
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1 vote
4 answers
61 views

Linear independence proof for set of functions

How do I prove that this subset of real valued functions $\{x, \sin{(x)}, \sin{(2x)}\}$ is linearly independent. Here is the proof suggest in the book: Consider the relationship $c_1 . 1 + c_2 . \sin{(...
user763322's user avatar
0 votes
3 answers
71 views

How to show that a function $f(x,y) = x^2-3xy+5y^2$ is coercive?

Definition. A continuous function $f(x)$, defined on $\mathbb{R^n}$, is coercive if: $$\lim_{||x||\xrightarrow[]{}\infty}f(x)=+\infty$$ Problem. Is the following solution is correct: $f(x,y)=x^2-3xy+...
36n's user avatar
  • 11
0 votes
0 answers
50 views

Can more functions be found?

In the book Discrete Mathematics with Applications Fifth Edition - Susanna S. Epp, section 7.1, problem: 7.1.4. b. Final all functions from $X=\{a,b,c\}$ to $Y=\{u\}$. Finding the pairs, $X \times Y = ...
Alix Blaine's user avatar
-1 votes
0 answers
44 views

Range of Reciprocal of |x|

My professor asked us to find the range of $\frac{1}{|x|}$ given that $x\in[-1,4]$. I sketched a graph of the equation, and assumed it must be equal to $[\frac{1}{4},∞]$ but I was told my answer was ...
Schrödinger's Cat's user avatar
1 vote
1 answer
44 views

$(-a)^x$ versus $-(a^x)$ help

$(-2)^3=-8$ and $(-2)^2=4$, right? And $-(2^3)=-8$ and $-(2^2)=-4$. So that means $(-a)^x$ does not equal $-(a^x)$. My question is why do we never see graphs of $(-a)^x$ then?? I tried graphing $(-2)^...
vergevoyage's user avatar
-2 votes
0 answers
63 views

Two different functions with a constant difference integral between them [closed]

Let $f$ and $g$ be any real-valued functions defined on $[0,L]$, and let their integrals be $F(x) = \int_0^x f(x') dx'$ and $G(x) = \int_0^x g(x') dx'$. For all $x \in [0, L]$, suppose $F(x) + \frac{\...
userflux9674's user avatar
0 votes
1 answer
53 views

Demonstrate that A is a countable set.

Question: Let $A = \{x \in \Bbb N \mid \exists y(x = 2y \lor x = y^2)\}$. Construct a surjective mapping $ f: \mathbb{N} \rightarrow A $. By doing this, you demonstrate that A is a countable set. ...
peterparker321's user avatar
0 votes
0 answers
20 views

Degree one meromorphic $\mathbb{C}\to \mathbb{C}$ mapping which is doubly quasi-periodic?

I am interested in functions $f$ which are meromorphic (depend only on $z$) and are almost (or quasi) periodic in the sense $$f(z+1) = e^{i\phi(z)} f(z),\qquad f(z+\tau) = e^{i\psi(z)}f(z)$$ where $\...
N Paul's user avatar
  • 1
1 vote
1 answer
68 views

Find the limit of the recursive sequence using the associated function

I have to find the limit of the following recursive sequence: $x_0=0; x_{n+1}=2x_n^3-x_n+\frac{5}{3\sqrt{6}}$ To study the monotony of the sequence I have considered that $x_{n+1}-x_{n}\geq 0 \iff f(...
axi's user avatar
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0 votes
0 answers
39 views

Show that the interval ⟨2, 5⟩ ⊆ ℝ⁺ is an uncountable set.

Question: Show that the interval $⟨2, 5⟩ ⊆ ℝ⁺$ is an uncountable set.\ To show that the interval $ \langle 2, 5 \rangle \subseteq \mathbb{R}^+ $ is an uncountable set, we can use Cantor's diagonal ...
peterparker321's user avatar
0 votes
1 answer
83 views

Computing the 21st derivative of $f(x) = \prod\limits_{n=1}^{5} \frac{x^n}{1-x^n}$ at $x=0$

Given a function, $f(x) = \prod\limits_{n=1}^5\frac{x^n}{1-x^n}$ Compute the value of 21st derivative of $f(x)$ at $x=0$ The answer given is $\frac{10}{21!}$. I proceeded with taking logarithm both ...
OpateItZOpatoOpate's user avatar
0 votes
1 answer
37 views

Maximum and Minimum of a cubic function

Maximum value of function $y = x^3-5x^2+2$ a) 5 b) $\infty$ c) 2 d) -5 We know to find maximum value of a function we take first derivative of the function and make it zero and get some point. And ...
user342326's user avatar
-3 votes
0 answers
33 views

How can you calculate baseball statistics over a given number of outs, given percentage per batter faced chances? [closed]

The givens are the BABIP (Batting average per Ball in play) - which describes the percent chance of a 'hit' (Which includes singles, doubles and triples, but not home runs, as those are not 'balls in ...
Brian Bunker's user avatar

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