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

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2answers
19 views

An injection from R × {0, 1} to R

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
0
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0answers
6 views

A property of Quasiconvex functions.

Let f be a strictly quasiconvex differentiable function and Df denote its gradient. Is the following implication true? :"Whenever f(y) < f(x), we also have (Df(x))'(y - x) < 0" . Suppose that f ...
0
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0answers
9 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
0
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0answers
8 views

how to show a concave function on discrete domain increases in x?

Define $g(x)=\frac{f(x)/x}{f(x)-f(x-1)}$ where x$\in$ $\mathbb{Z}$. Known that $f(x)$ has the concave extension in every consecutive $x$, i.e: $f(x+1)+f(x-1)-2f(x)<0$ holds $\forall x$. My question ...
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0answers
20 views

Consider the integral expression in $x$ [on hold]

$P(x)=x^3+x^3+ax+1$ where $a$ is a rational number. At $a=3$ the value $P(x)$ is a rational number for any $x$ which satisfies the equation $x^2+2x-2=0$, and in this case the value of $P$ is $12$.
4
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1answer
67 views

If $f(f(x))=x$ does that mean $f(x)$ equals its inverse?

Given any real function, if $f(f(x))=x$ does that mean $f(x)$ is its own inverse? I am confused since $f^{-1}(f(x))=x$ and this is a fact, so can we assume that $f(x)$ will equal $f^{-1}(x)$ by ...
0
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2answers
28 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
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0answers
11 views

Existence of subgradient of a quasiconvex function

Does a continuous quasiconvex function always have a subgradient? More strongly, is it true that if $f$ is a continuous quasiconvex function, then for each $x$ there is a vector $c$, such that for ...
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0answers
6 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
0
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0answers
27 views

Function that maps all vectors to the origin?

I need a function that will map any vector (and any point on that vector) in the cartesian plane to (0,0) using only addition and subtraction. Is this possible?
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0answers
9 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
0
votes
1answer
18 views

Functions of modulus

How do I calculate the range of any modulus function? I know that if $x <2$ then it's expansion is negative and if $x>2$, it's expansion is negative, but will it help? Consider an example, $$f ...
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votes
4answers
44 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
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votes
2answers
49 views

What is n (Natural number) if the function has to have a limit not equal to zero or infinite? [on hold]

$$\lim_{x \to 0} \frac{(\tan(x))^n - x^n}{x^6}$$ What is n (Natural number) if the function has to have a limit not equal to zero or infinite?
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0answers
17 views

Mathematical formal expression of find “subfunction” in function

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...
3
votes
8answers
81 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
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votes
2answers
45 views

Is $(g \circ f)^{-1}$ equivalent to $(f^{-1} \circ g^{-1})$? Why? [on hold]

Is the inverse of composition of function is equivalent to composition of inverse functions? Why?
1
vote
1answer
20 views

Value of the difference if functions

Is there a relatively simple function $f(x)$ such that $f(x)-f(x-1)=x^n$? Note that $n$ is a positive integer. Thanks so much!
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3answers
50 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
2
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2answers
24 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
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1answer
48 views

stuck on an function question [on hold]

I'm studying CompSci, While I'm having fun with that, I haven't had a higher level math class since 2009. Unfortunately, I'm required to take calculus in order to pursue something I'm passionate ...
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votes
0answers
31 views

One-to-One functions help [duplicate]

The one-to-one functions $g$ and $h$ are defined as follows: $$g=\{(9,8), (5,9), (8,-9), (9,-2)\}$$ $$h(x)=4x-9$$ SOLVE $$(g^{-1})(-9)=?$$ $$(h^{-1})(x)=?$$ $$(H o H^{-1})(7)=?$$
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1answer
40 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ 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 ...
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0answers
52 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
7 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
31 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
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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?
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votes
1answer
44 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
66 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
27 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? ...
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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
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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
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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 ...
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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 ...
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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
123 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
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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 ...
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2answers
39 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 ...
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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|) = ...
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2answers
50 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 ...
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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?
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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 ...
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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 ...
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0answers
28 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 $f(6)$?
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1answer
42 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} \\ $ ...
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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 ...
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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 ...
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1answer
48 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 ...