Elementary questions about functions, notation, properties, and operations such as function composition.

learn more… | top users | synonyms (1)

0
votes
1answer
11 views

These maps from the components into a directed system are injective when the directed system maps are.

Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there ...
0
votes
1answer
29 views

What are the composite functions

f : $\mathbb{R} \to \mathbb{R}$ $$g(x)=\begin{cases} \frac1n,&x\in\Bbb Q\text{ and }x=\frac{1}n\text{ in lowest terms}\\ \sqrt{2},&x=0\ \end{cases}$$ g(x) is the inverse of f(x) determine ...
0
votes
1answer
34 views

Is the function invertible?

$$f(x)=\begin{cases} \frac1q,&x\in\Bbb Q\text{ and }x=\frac{p}q\text{ in lowest terms}\\ 0,&x\notin\Bbb Q\;. \end{cases}$$ Is the function $f|_D$ invertible? If so, describe its inverse ...
0
votes
2answers
38 views

What is the domain of the function

I think the subset D is 1/n where n is an element of natural numbers. Can someone help me with this, thanks in advance
8
votes
4answers
125 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
1
vote
1answer
33 views

what does this phrase mean?

Can someone explain in everyday language what "monotonically" and "quadrilaterally" mean in the following sentence? Muscular torques increase monotonically with increasing diameter for lids with ...
0
votes
2answers
41 views

From the graph find the number of solutions.

The figure below shows the function $f(x)$ . How many solutions does the equation $f(f(x))=15$ have ? $a.)\ 5 \\ b.)\ 6 \\ c.)\ 7 \\ d.)\ 8 \\ \color{green}{e.) \ \text{cannot be determined from ...
1
vote
1answer
54 views

Existence of $x_0$ such that $f(|x_0 + a|) = f(|x_0|)$ given $f \colon \mathbb R \to \mathbb R$ and $a$

So I have this function $f : \mathbb{R} \to \mathbb{R}$ that is continuous and I have $a\in\mathbb{R}$. I have to prove that exists an $x_{0}\in\mathbb{R}$ such that this works: $$f(|x_{0}+a|) = ...
0
votes
2answers
51 views

Integral with only a list of values

I am supposed to perform an integral of function $y(x)$ from say $x_1$ to $x_2$. Now the issue is I don't have an actual function $y(x)$, but I do have a list of values for $y$ and $x$. In what way ...
2
votes
2answers
37 views

Are differentiable and strictly decreasing functions always concave?

If a demand function is continuously differentiable and strictly decreasing in price, does that mean it will be always concave?
-4
votes
0answers
38 views

Map 1 to 1 and 0 to -1. [on hold]

I have a really simple question, but I cannot find an answer. Suppose that I have an input number $x$. And its domain is $1$ and $0$. Is it possible, that when $x$ is $1$, map it to $1$, but ...
0
votes
3answers
45 views

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$?

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$? I am a high school student and stuck at this simple question on domains and ranges of functions. I have done the ...
-2
votes
0answers
31 views

suppose $f(x)$ is a function satisfying a specific relation [on hold]

if $f(x)$ is a function such that : $$\left\{\begin{matrix} f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x}) & \\ f(4)=65& \end{matrix}\right. $$ what's the value if $f(6)$?
0
votes
1answer
43 views

The value of $x$ for which function attains max value

At what value of $x,\ x\in \mathbb{Z}$ will the function $\dfrac{x^2+3x+1}{x^2-3x+1}$ attain its maximum value . $\color{green}{a.)\ 3 }\\ b.)\ 4 \\ c.) -3 \\ d.)\ \text{none of these} \\ $ ...
1
vote
2answers
26 views

Is the co-domain needed if we have the range? [duplicate]

Why do we need the co-domain if we have the range? I know what both mean. Isn't it just better to use the range instead of the co-domain when defining a function? This question brought up to me when ...
1
vote
0answers
11 views

Equation for adjusting a scalar, trouble with fractional values

I have a real number - call it $s$, that I use to scale other numbers. Think $2\times$ scale $1/2\!\!\times$ scale, things of that nature (I am no math expert so please bear with my elementary ...
0
votes
2answers
23 views

partial functions basics

$f: \mathbb{Z} \to\mathbb{N}$ is defined as $$ f(x)= \begin{cases} 2x-1, & \text{$x \gt 0$} \\ -2x, & \text{$x \le 0$} \end{cases} $$ one to one proof f is onto proof ...
1
vote
1answer
49 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
-2
votes
0answers
55 views

How many increasing functions $f:\{1,\ldots,n\} \to \{1,2,\ldots,n\}$ are there such that $f(i) \ge i , \forall i=1(1)n$ , where $n \in \mathbb N$?

Let $n\in \mathbb N , n \ge 3$ . How many increasing functions $f:\{1,,\ldots,n\} \to \{1,2,\ldots,n\}$ (i.e. $f(i) \ge f(j) , \forall i=1(1)n$ ) are there such that $f(i) \ge i , \forall i=1(1)n$ ?
0
votes
1answer
83 views

The limit of $\sin \lfloor x\rfloor/\lfloor x\rfloor$ as $x\to 0$

If $$f(x) = \begin{cases}\dfrac{\sin \lfloor x\rfloor}{\lfloor x\rfloor} &, \lfloor x \rfloor \neq 0 \\ \quad 0 &, \lfloor x\rfloor = 0. \end{cases}$$ Find limit of $f(x)$ when $x$ tends to ...
0
votes
1answer
24 views

What is the subset D of the domain

What is a subset $D$ of the domain of $f$ such that $f\rvert_D$ is simultaneously one-to-one and onto the range of $f$? The function $f: \mathbb{R} \to \mathbb{R}$ is given as $$ f(x) = ...
0
votes
1answer
11 views

What is the range of the function

let f:R->R What is the range of the function f I think it is(-infinity to infinity). But i am confused because p/q is in their lowest term. Can Someone please help me, Thanks in advance
0
votes
1answer
33 views

Find inverse $f^{-1}$ of a function $f(x,y)=(x-y,x-10y)$ [duplicate]

I know how to find inverse function if the given function is in the explicit form. Could someone show on this example how to find $f^{-1}$? Thanks for replies.
0
votes
3answers
31 views

Examine if function $f:\mathbb{R^2}\rightarrow \mathbb{R^2}$ which is defined as $f(x,y)=(2x-y,x-4y)$ is bijective. If bijective, find $f^{-1}$.

Function is bijective when it is injective and surjective. Function is injective if $$(\forall x_1,x_2 \in A)f(x_1)=f(x_2)\Rightarrow x_1=x_2$$ and surjective if $$(\forall y \in B)(\exists x \in ...
1
vote
1answer
42 views

About integration by substitution

I know how the method goes: we want to find $\int{f(g(x))g'(x)dx}$, which by the reverse chain rule equals $\int{f(u)du}$. My (maybe stupid) question comes from the integrals with the form ...
0
votes
0answers
22 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
3
votes
1answer
37 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
0
votes
1answer
23 views

Quick question on a geometric translation.

I am following the text Advanced Calculus by Lynn H. Loomis and Shlomo Sternberg. In this passage why is $y = f(x)$ iff $s = f(a+t) -f(a)$?
2
votes
0answers
17 views

Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
1
vote
2answers
36 views

a theory of transcendental functions?

Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus ...
0
votes
4answers
38 views

How to explain that function has positive and negative values around zero?

I have following function $$f(x)=\begin{cases} x^2\cos\left(\frac1x\right) &\text{if }x\neq0\\ 0 &\text{if }x=0 \end{cases}$$ How can I prove that this function in every area of zero has ...
18
votes
3answers
944 views

Does there exist a continuous function from [0,1] to R that has uncountably many local maxima?

Does there exist a continuous function from $[0,1]$ to $R$ that has uncountably many strict local maxima?
1
vote
1answer
49 views

What is the significance of squaring a number? [closed]

I've always been baffled by the significance of squaring a number. I understand what it means ( $10^2 = 10 \cdot 10$) but what is the significance of doing this? Obvious examples are: $E=mc^2$ Area ...
1
vote
2answers
64 views

I am trying to find the limit of P(x)

When I am looking for a $\lim\limits_{x \to -1} P(x)$ where P(x)$= \sum \limits_{n=1}^\infty \left( \arctan \frac{1}{\sqrt{n+1}} - \arctan \frac{1}{\sqrt{n+x}}\right) $ do I have to ignore a ...
4
votes
2answers
57 views

Study this function $f(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$

I need to study this function: $$f(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$$ and I need to show Max and Min point. The first thing is define the Domain, so: $$\left\{\begin{matrix} \sqrt[3]{x-1} > ...
0
votes
1answer
30 views

Why does a line integral not depend on the parametrization you use?

I have a question about my calculus course: Why is it true that a line integral over a certain functiondoes not depend on the parametrization you use?. For example, take a function $f(x,y,z)$ of 3 ...
3
votes
0answers
52 views

Looking for a bound on a function involving $\sinh$

Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$). Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of ...
0
votes
1answer
5 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
1
vote
0answers
48 views

Determine null, extreme and inflection points of function $f(x)=\frac{x+e^x}{x-e^x}$

This function has a null point, but I can't compute it from equation $f(x)=0$ which gives $$\frac{x+e^x}{x-e^x}=0$$ $$x+e^x=0$$ How to compute this equation? Extreme points can be computed from ...
0
votes
1answer
34 views

There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
1
vote
1answer
27 views

Drawing a graph of a function.

$h_{1}=pq-\frac{1}{2}kq^{2},\ h_{2}=pq-kq^{2}, \frac{dh_{2}}{dh_{1}}=\frac{p-2kq}{p-kq}$, $k,p$ are constant. My question are how can I draw a graph of function $h_{2}$ measuring $h_{1}$ on the ...
8
votes
4answers
141 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
-2
votes
0answers
42 views

What is the relationship between these numbers? [closed]

Please would you help with this? Thank you in advance. Numbers are grouped into 3 inputs, giving 3 outputs. I am trying to establish the connection between the numbers. Each group of 3 appears to ...
2
votes
1answer
37 views

Equality between functions?

Given are real functions as follows: $f_{1}(x)=x, \; f_{2}(x)=\frac{x^2}{x}, \; f_{3}(x)=\sqrt{x^2}, \; f_{4}(x)=\left (\sqrt{x} \right )^2$ Are there any equal among them? I checked the domains ...
0
votes
0answers
7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
2
votes
1answer
27 views

Why not an Absolute maximum in an open interval?

The function $x^3+x^2\: \text{has a maximun value at}\: x=-\frac{2}{3} \text{in (-1, 0) }.$ My question is why call it a Local Maximun and not an Absolute Maximum when it is the highest value in that ...
0
votes
1answer
33 views

A question on vector subspace [duplicate]

Let $V$ be the vector space of all functions $f \colon \mathbb{R} \to \mathbb{R}$ over $\mathbb{R}$, is the set of functions which are continuous a subspace? I think if you add functions which are ...
3
votes
0answers
46 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
3
votes
1answer
17 views

What is the difference between functions and operations?

Wikipedia says that an operation $\omega$ is a function of the form $\omega: V \to Y$, where $V \subset X_1 \times\cdots\times X_k$. But as far as I know, every function's domain is a set, so ...
1
vote
1answer
20 views

Onto (surjective) functions of 2 variables [closed]

I have a couple of functions I'm curious about: $f(m,n)=m^2 -n^2$ and $f(m,n)=|m|-|n| $, for $m,n\in \mathbb{Z} $. The codomain also consists of all integers. My understanding is that for this ...