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

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0
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1answer
10 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
1
vote
2answers
19 views

Is my solution correct for inverting the function $a/(n-b)$

I inverted the following function: $$r(n) = {a \over n-b}$$ Inverting the function: $$r^{-1}(n) = { a + bn \over n}$$ Is my solution correct?
-4
votes
1answer
52 views

math function to generate this series: 25, 35, 42, 47… [closed]

I think next should be 51 since the "difference of difference" the nr. decreases by 1 each step (35-25 = 10, 42-35 = 7, 47-42 = 5 so we have 10, 7, 5, next one should be 4)
0
votes
1answer
24 views

ideas on flexible function to describe this data?

I am trying to think of as simple as possible of a function that could to do a decent job at fitting the curves in the figure below. I have tried various sigmoid functions (e.g., logistic), a simple ...
0
votes
2answers
15 views

Set inequality with functions

Let $f:A \to B$ with $X\subset A$ and $Y \subset B$. I'm trying to prove $X \subset f^{-1}(Y) \implies f(X) \subset Y$. Note that $f^{-1}(Y)$ denote the inverse image of $Y$. I've been element ...
0
votes
2answers
27 views

How to create an explicit formula of a function

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a differentiable and nondecreasing function such that $\phi(x)=-1$ for $x<-1$, $\phi(x)=1$ for $x>1$, and $\phi^\prime(x)>0$ for $x\in(-1,1)$. ...
5
votes
3answers
87 views

Standard Indicator function?

Is there any standard function that outputs $1$ if an inequality is true and $0$ otherwise? e.g $$F = \begin{cases} 1 & \text{if }x>D\\ 0 & \text{otherwise} \end{cases}$$
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2answers
46 views

Limits and Functions [closed]

Give an example of a function, $f(x)$, so that $\lim\limits_{x \to 1} f(x)$ does not exist but $f(1)$ is defined
1
vote
1answer
29 views

Conformal mapping of two annuli to the punctured unit disc

What is the general procedure for finding a holomorphic bijection from the region $ \Omega = \{z \in \mathbb{C}: |z - a| > 1, |z + a| > 1 \}$ to the punctured unit disc?
-5
votes
0answers
13 views

Solve for marginal prbability density function [closed]

Hello everybody, can someone help me with the first question How do i know the integration is from where and to where ?, since x and y are dependent ? Thank you
0
votes
1answer
17 views

Finding the equation for a cagr with yearly production

I have this scenario: Miners produce $500$ per year Scientists improve returns by $0.04\%$ per year Calculate the return in 20 years for an amount M of miners $(1-10)$ and $S$ of scientists ...
1
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2answers
52 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
2
votes
1answer
47 views

Prove that the solution of $y'+y=\arctan(e^x), y(0)=2$ admits horizontal asymptote.

Let us consider the Cauchy problem: $$y'+y=\arctan(e^x),\ \ \ \ y(0)=2$$ Prove that the function $y(x)$ admits horizontal asymptote without solving the problem.
1
vote
0answers
23 views

How to prove the equivalence between sets?

I want to prove that if $A$ is equivalent to $C$ and $B$ equivalent to $D$ then $A\times B$ is equivalent to $C\times D$. I used $f(a,b)= (h(a),g(b))$ where $h: A \rightarrow C$ one-to-one ...
10
votes
0answers
126 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
0
votes
0answers
22 views

Show that the identity function over any set is a bijection

Let $A$ be any set, and let $I: A\to A$ be the identity function on $A$. To show this identity function over $A$ is a bijection. We can show that it is injective and surjective. We can readily know ...
3
votes
3answers
83 views

Given $Re\{f'(z)\}$, to find $Im\{f(z)\}$

An analytic function $f(z)$ is such that $\Re\{f'(z)\} =2y$ and $f(1+i)=2$. Then the imaginary part of $f(z)$ is $-2xy$ $x^2-y^2$ $2xy$ $y^2-x^2$ Here, by using Milne Thomson's ...
1
vote
1answer
26 views

Can we find two different real functions f and g such that f is a composition of g's and g is a composition of f's?

Can we find two different real functions $f$ and $g$ such that $f$ is a composition of $g$'s and $g$ is a composition of $f$'s? ($f,g \; \mathbb{R} \rightarrow \mathbb{R})$. My ideas: Well, if f is a ...
2
votes
1answer
16 views

For any strictly increasing convergent sequence $x_n$, the sequence $f(x_n)$ is convergent

Let $f(x)$ be defined on R and be strictly increasing. Claim: for any strictly increasing convergent sequence $x_n$, the sequence $f(x_n)$ is convergent. I believe it's false. Think about ...
1
vote
1answer
36 views

What kind of function may satisfy this set of properties?

What's an explicit, expressible function that I can program on a computer using basic arithmetic that satisfies the following properties: $$ f \text{ is differentiable}\\ f(0)=0\\ f(1)=1\\ ...
0
votes
0answers
36 views

Prove that if a function f is continuous then it is monotonic

Let ($L_1 , \le_1 $) and ($L_2 , \le_2 $) be two posets. A function $f : L_1 \to L_2 $ is continuous if $ \forall E_1 \subseteq L_1, E_1 \neq \emptyset, f(LUB(E_1)) = LUB(f(E_1)) $ Prove that if $f$ ...
0
votes
1answer
29 views

understanding of Functions [closed]

I Have this question im tring to get my head around. Say i need to find L(abaa) and L(aaa). What are the steps and process for this. I don't understand with the elements in the strings. Any guidence ...
1
vote
5answers
61 views

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Does being onto guarantee the sets are finite?

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Determine which of the following statements are true: If $A$ is finite then $B$ is finite. If $B$ is finite, ...
0
votes
0answers
13 views

Find the interquartile range of this piecewise function

$$f(t)= \begin{cases} |2-t|, & \text{$1 \leq t \leq 3$} \\ 0, & \text{Otherwise} \end{cases}$$ I have graphed the piecewise function, but I have no idea how to find its interquartile range! ...
1
vote
1answer
25 views

Prove that if $X$ and $Y$ are independent discrete variables, then $f(X)$ and $f(Y)$ are independent.

Prove that if $X$ and $Y$ are independent discrete variables, for $f: \mathbb{R} \rightarrow \mathbb{R}$, then $f(X)$ and $f(Y)$ are independent. Here is the exact same question. I define ...
2
votes
3answers
33 views

Are the sum and/or product of two increasing functions also increasing?

Question: Let $f(x)$ and $g(x)$ be two increasing functions. a) Show that their sum is also increasing. b) Investigate the corresponding claim for the product of two increasing functions. ...
-1
votes
0answers
40 views

On some counterexamples about continuity, intermediate value propriety, Riemann integrability, and antiderivatives

At school, I have been studying the relationship between continuity, monotonicity, and Riemann integrability. In doing so, I tried to make up some examples and counterexamples, but there are some ...
1
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0answers
12 views

Determining the Domain and Range of a multi-dimensional function

$ f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z} $ i) Write down the domain of $f$ ii) Determine the range $T$ of $f$. For each $c \in T$ find a point $(x,y,z) \in \mathbb{R}^3$ such that $f(x,y,z) = 1 $. ...
2
votes
1answer
56 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
0
votes
2answers
52 views

Is the following function finite?

Let $1 < p < 2$ and $\forall \ \xi \in \mathbb{R}-\{0\}$ sucht that $$\varphi (\xi) = \frac{|1 + \xi|^p - 1 - p \xi}{|\xi|^p}.$$ We have that $$\lim_{\xi \to \pm \infty} \varphi (\xi) = 1 \;\; ; ...
1
vote
1answer
12 views

If the period of the function $cos(nx)sin(5x/n)$ is $3\pi$ then what should be number of integral values of n?

If the period of the function $\cos(nx)\sin(5x/n)$ is $3\pi$ then what should be number of integral values of $n$ ? My approach : I tried like period of $\cos(nx)$ is $2\pi$/n and $\sin(5x/n)$ is ...
2
votes
1answer
30 views

Homomorphisms Groups Kernel

Given G: group of units in Z mod 14 under multiplication. A function sends the integers under addition to G. $f(n)$ = $[3]^n$ I am just checking whether I am correct in stating that the kernel ...
0
votes
0answers
21 views

Functions understanding

So lets say im Trying to find (G o F)(7) and (F o G) I want to see if im on the right track G o F = f(g(7)) = f(7-1) = (7-1)^3 = 261 F o G = g(f(7)) ...
3
votes
1answer
68 views

Does there exist a continuous onto function $f:[0,1] \to (0,1)$?

If there is, what's an example. If not, how do I prove none exists?
2
votes
1answer
23 views

How could I show that the set of degenerate critical points of a $C^{\infty}$ function is a closed subset of $\mathbb{R^n}$?

What would be a good idea to show( or finding a counter example) the set of degenerate critical points of a $C^{\infty}$ function $f: \mathbb{R^n}\to\mathbb{R}$ is a closed subset of $\mathbb{R^n}$? ...
1
vote
3answers
80 views

Solve for $n$ in $2^n=8$

So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. ...
2
votes
1answer
51 views

Inverse of $f(x) = 3x + \cos(x)$

Was hoping someone could help me find the inverse of $f(x) = 3x + \cos(x)$ The steps I took were: $y = 3x + \cos(x)$ $x = 3y + \cos(y)$ $x - 3y = \cos(y)$ $\arccos(x-3y) = y $ But I still have a ...
0
votes
0answers
29 views

Intuition between convex function and convex set

In my text (Luenberger), there is a proposition about convex set Prop: let $f$ be convex function on convex set $\Omega$. The set $\Gamma_c = \{{x: x\in \Omega, f(x) \leq c}\}$ is convex ...
1
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2answers
57 views

What is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ and } f(\mathbf y) - f(\mathbf x)$?

The Problem: Let $f(\mathbf x)$ be a convex function on $\mathbb R^n$. Given two points $\mathbf x$ and $\mathbf y$, what is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ ...
2
votes
2answers
43 views

Showing that a function is surjective (onto)?

For example : $F:\Bbb R\rightarrow\Bbb R$ defined by $F(x) = \frac{2x+1}{3}$ I let $F(x)=Y$ which gives $Y=\frac{2x+1}{3}$ then simplify and solve for $x$ , what I have at the end is ...
-1
votes
1answer
16 views

Proving the limit of the power of two functions is the power of the limits?

I've already seen a couple of times both on questions here (like Value of $\lim_{n\to\infty}{(1+\frac{2n^2+\cos{n}}{n^3+n})^n}$ or Problem of limit of power function) and in other online resources ...
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votes
0answers
16 views

X simple and Y simple problem with double integrals

Let T be the triangle in the plane with vertices (−1, −1), (1, 0), and (1, 3). Compute Double integral 8x^3y dA I understand that you first determine whether the problem is an X simple or Y simple or ...
1
vote
1answer
16 views

What does the notation $\{ \pm 1 \}^X$ in relation to functions and hypothesis classes means in the context of PAC learning over half spaces?

I was reading the following paper (on PAC learning over half-spaces) and encountered the following notation for a hypothesis class (on page 4): $$\mathcal{H} \subset \{ \pm 1 \}^X$$ However, it was ...
7
votes
5answers
72 views

Show that if $g \circ f$ is injective, then so is $f$.

The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...
0
votes
2answers
108 views

$f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$

Let f be defined from real to real $f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$ Now how to find the period of this function f(x)? Can someone provide me a purely ...
0
votes
0answers
12 views

Characterization of monovalued functions

Let $f$ be a binary relation. Let $(\bigcap G)\circ f = \bigcap_{g\in G}(g\circ f)$ for every set $G$ of binary relations. Can we prove that $f$ is monovalued (a function)?
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votes
2answers
24 views

Limit of mappings over a real function F

I have read in a book that if we let $F$ be a bounded function, and $M$ a mapping, such that $F = MF$ is satisfied. Then $$F = \lim_{N \to \infty}M^NF$$ is valid. Can this be the case in general? ...
1
vote
1answer
36 views

Injections from a set of functions to R

Show there is an injection from $\Bbb R^2 \to \Bbb R $ Does there exist an injection from $X \to \Bbb R$ where $X $ is the set of all functions where f(x)=x for all but finitely many x. This is a ...
5
votes
4answers
328 views

Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$ for a 4-th degree monic polynomial

If $f(x)=x^4+ax^3+bx^2+cx+d$. Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$. This problem has troubled me a lot.The more I try to solve it,it becomes lengthier. My problem is that there are four ...
0
votes
1answer
53 views

Find 2 functions

I need to find two functions or rather two sequences, both of which are borderless but actually converge if you look at $min(a_n, b_n)$. The min has a limes but neither of them does for itself and min ...