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

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0
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2answers
18 views

Find the value of this $2f\left ( \frac{1}{2} \right )$

IF $f(f(x))=1-x$, Find $$2f\left ( \frac{1}{2} \right )=??$$ help guys, I really tried but I couldn't.
0
votes
0answers
10 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,R))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le \frac{4}{R}$ for ...
0
votes
2answers
75 views

If $f'(0)\ge0$ how to find $f(10)$?

A real valued function satisfies the condition: $f(x+y^3)=f(x)+[f(y)]^3$ for all real $x$, $y$. If $f'(0)\ge0$ how to find $f(10)$ ?
0
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0answers
40 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
0
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0answers
3 views

Creating an evenly distributed function $B=B(p)$ over the range $p=p_{min}$ to $p=p_{max}$

In some notes on statistical thermodynamics, I encountered this: The momentum distribution function $B(p)$ is evenly distributed over the allowed range: ...
1
vote
0answers
29 views

What is the opposite of a derangement?

A derangement is a bijection $f : A \rightarrow A$ such that $f(x) \ne x$ for all $x \in A$. Is there a name for a bijection $f : A \rightarrow A$ that is not a derangement? That is, is there a name ...
1
vote
1answer
20 views

Differentiating composite function

Can anyone say the basic formula for the differentiation of the composite functions? Is it similar to chain rule?
-1
votes
1answer
43 views

How can I prove that this function is continuous in (0,0)?

I have this function: $$ \lim_{(x,y)\to (0,0)} = \frac{2(1-\cos(xy))+\arctan(x^4)-x^2(x^2+y^2)}{(x^2+y^2)^\alpha} $$ I have to find which $ \alpha$ makes the function continuous. But my first problem ...
1
vote
1answer
23 views

Find bijective correspondence between the sets

Find bijective correspondence between the set of all functions of $X$ in the set $\left\{ 0,1 \right\}$ and the power set of set $X$ and find $| 2 ^ X |$, if $| X | = n.$ My thoughts: ...
4
votes
1answer
63 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
votes
1answer
26 views

Need a function for the following…

I'm trying to come up with a function formula for $y$, that is a broad curve that passes through $(0,0)$ and almost crosses $(100, y)$ but never does. ( so $x < 100$) Can you please help me? ...
0
votes
2answers
27 views

How do I reverse the smooth-step equation?

I'm using the "smooth step" equation for an easing curve: $y = 3x^2 - 2x^3$ I would like to reverse this equation so that given y, I can find ...
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
32 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
37 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
122 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
36 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
53 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
49 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
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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
44 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
27 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
41 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
25 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
10 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
45 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
54 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
81 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
32 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
28 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
40 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 ...
16
votes
3answers
930 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? [on hold]

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
29 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 ...