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

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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. [closed]

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
47 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 ...
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
27 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
52 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
44 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
19 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
39 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
46 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 ...
19
votes
3answers
951 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
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
6 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
143 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
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
48 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 ...
1
vote
1answer
15 views

Is it possible to express the indicator function for a real interval in terms of other function/s?

Suppose that I've an indicator function defined for an interval in $\mathbb R$, i.e., suppose that $f(x)=1$ if $x\in(a,b)$ and $f(x)=0$ otherwise. Then can I express $f$ without using indicator ...
1
vote
2answers
74 views

Is 'clamp' a formally recognized mathematical function?

I was surprised to find the clamp function absent from Mathworld and Wikipedia. Is this mainly a concept particular to computer programming? Is it known by another ...
0
votes
2answers
29 views

Is the composing of functions always commutative?

I have a question for my math study. It seems quite simple, but I just can't find a counterexample for the following: The composition of two functions is always commutative Could you help me with ...
0
votes
1answer
34 views

Why is the following true? functions

$$x , x_0 \in [a,b]$$ $x_0$-fixed $f \in D(a,b)$- differentiable on [a,b] $$\triangle (x)=f(x)-f(x_0)-f'(x_0)(x-x_0)$$ $$\triangle '(x)=f'(x)-f'(x_0)$$
4
votes
0answers
75 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
0
votes
4answers
64 views

If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one?

The title says it all - but to reiterate: If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one? I think not. Anyone have a good proof for this? This is simply ...
2
votes
3answers
24 views

Function and Domain with Deleted Neighbour - Beginner Question

I have a simple question about functions and domains. Consider the following function: $$f(x) = \frac{ x^2-9}{x-3}$$ I often see in the textbooks mentioning that the domain of this function can be ...
0
votes
1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
0
votes
1answer
60 views

Inverse function $g^{-1}$

The function $g$ is defined by $$g(x)= 3-2x-4x^2, x\in \mathbb{R},x\leq -\frac{1}{4} $$ Find the inverse function $g^{-1}$. Calculate the value of $x$ for which $g(x)=g^{-1}x$. My attempt, ...
0
votes
0answers
14 views

Extension of a quasiconvex function

A function $f$ defined on a subset $S$ of the $n$ dimensional Euclidean space is said to be quasiconvex if $f(ax + (1-a)y) \le\max \{f(x) , f(y)\}$ for all $x, y \in S, a \in [0,1]$. Now, suppose ...
0
votes
1answer
7 views

Convex combination of quasiconvex functions.

Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?
1
vote
2answers
22 views

Product of two continuous, non-negative and monotone non-decreasing function is itself..

My question is simple: is the product of two continuous, non-negative and monotone non-decreasing function itself a continuous, non-negative and monotone non-decreasing function? I believe the ...