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

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9 views

Find the fixed points of a function $f(x) := exp(x - 2)$ using a recursive algorithm

I need to find the fixed points (i.e. when $f(x) = x$) of the following function $f(x) := exp(x - 2)$. I understood that the fixed points should be the intersecation points between $f(x)$ and a ...
1
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1answer
21 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
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0answers
8 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
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1answer
21 views

Inverse function.

A function $h$ is defined by $h:x\rightarrow 2-\frac{a}{x}$, where $x\neq 0$ and $a$ is a constant. Given $\frac{1}{2}h^2(2)+h^{-1}(-1)=-1$, find the possible values of $a$. Can someone give me some ...
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1answer
23 views

Quick question about mod equation

So here is the mod function: $$5 ^ {31} \cdot 2 ^{789} - 23^{23}\pmod{10}$$ Is there a way to shorten it, or I must calculate it plain numbers? I have tried the mod powers rule, but except for the ...
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2answers
27 views

Showing that a function is strictly increasing

Let $f(x)=x/(1+|x|)$, $x\in\mathbb{R}$ This is a simple question but I am a bit stuck to show directly that $f$ is strictly increasing, so without any tools like the 1st derivative test, so just using ...
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0answers
12 views

Properties of a function needed for not having a zero

While studying ODE I thought of the following problem: Let $f:A\subset\mathbb{R}\to\mathbb{R}$ and $x_0\in A$ such that $f(x_0)=0$. What properties should have $f$ so as to allow us to conclude that ...
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1answer
13 views

Functions - Trig - Determine [on hold]

The vertical displacement of the end of a robot arm (in cm) at time t (in [16 marks] seconds) is given by y = 8 + 7 cos 3t + 7 cos 6t: (a) Find all times, t > 0, (in exact form i.e. in terms of ...
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1answer
18 views

Formula with differing output based on sign of input

I'm looking for a mathematical formula that will reproduce this pseudo-code: if x >= 0 x+=1 else x-=1 If this is possible, what would such a formula look ...
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1answer
14 views

Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
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1answer
53 views

Need Help Understanding Notation With Functions

Original picture: LaTeX approximation: $$f\color{blue}{\substack{(x)\\x\to\infty}}=\pm\sqrt{\frac{(x^2+x)^3}{\pi}}.$$ What does the notation highlighted in blue mean? I understand that ...
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3answers
56 views

Finding lower/upper bounds for $\prod_{i=2}^n \log(i)$

I have a homework problem where I need to asymptotically order a set of functions, and $\prod_{i=2}^n \log(i)$ is one of them. Is there a tight upper/lower bound for this function? I've tried the ...
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1answer
11 views

Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive.

Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive, where $F = \{ f \mid f : A \to A\}$ and $P = \{f\in F \mid f\text{ is one-to-one ...
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1answer
53 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
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1answer
16 views

Matlab noob looking for probably very basic advice to plot a curve [on hold]

Basically just started to use Matlab and am struggling with this basic example. My function or variable d is undefined but am unsure on how to progress. If anyone can guide me through the first ...
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0answers
17 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
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2answers
39 views

What's the relation between a fixed point and a root of a function?

A fixed point of a function $f$ should be an $x$ in the domain of $f$, such that $f(x) = x$. On the other hand, a root (or zero) of a function, should be an $x$ in the domain of $f$, where $f(x) = ...
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0answers
32 views

Find inverse of a double function

I have the following function: $$f(x)=\begin{cases}3x+1,~x\gt 0\\2-x^2,~x\leq 0\end{cases}$$ and I need to find its right inverse. So far I got that, ...
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0answers
27 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
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1answer
16 views

Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
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0answers
24 views

Implicit function theorem and derivatives

I'm a bit confused about determing $y'$ and $z'$ If I differentiate both equations wrt $x$ I get: $2x+2y\frac{dy}{dx}+2z\frac{dz}{dx}=0$ and $y+(x+z)\frac{dy}{dx}+y\frac{dz}{dx}=0$ Now because ...
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2answers
25 views

What does it mean to write $(y,z)=G(x)$

I understand this is a map from $\mathbb{R} \mapsto \mathbb{R^2}$. Is it the case that $y=y(x)$ and $z=z(x)$? i.e Are $y$ and $z$ individually functions of $x$ as well as being so jointly ...
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0answers
7 views

References for a notion related to radially lower semicontinuity

Let $E$ be a real vector space, $C\subset E$ be a nonempty convex set and $z\in C$. Let $f:C\rightarrow\mathbb{R}$ such that $$ \textbf{(A)} \quad f(z)\leq\limsup_{t\downarrow 0}f(z+t(w-z))\quad ...
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1answer
59 views

Can a function map $\mathbb{R}\mapsto\mathbb{R^2}$ [on hold]

Is the mapping in the title of this question possible?
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1answer
55 views

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 “straight” lines?

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 "straight" lines? Using https://www.desmos.com/calculator for plotting.
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2answers
52 views

Prove or disprove: If $\lim_{n\to\infty} (a_{2n} - a_n) =0$, then $a_n$ has a limit (not infinity). [duplicate]

I need to prove if this is true or false.If true then I need to Prove and if false I need to provide an example that disproves the statement.I tried many times but it didnt work. Note:- Im new so I ...
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0answers
13 views

How to design the analyticity of a function to find when the variable becomes zero

Suppose there is a n-arity function $f(x_0,x_1,...x_n)$ and I want to use it as fitness function. By optimizing function $f$, I want to know when variable $x_0$ or $x_1,...,$ or $x_n$ comes to zero. ...
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0answers
13 views

Properties Of Different Kinds Of Functions - Venn Diagram Method [on hold]

Suppose there are two sets A and B.A has m elements and B has n elements. What will be maximum number of 1)Functions 2)One-One Functions 3)Many-One Functions 4)Into Functions 5)Onto Functions ...
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1answer
23 views

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$]

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$] ? How should the graph be ?
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3answers
61 views

How do I prove this Limit? [on hold]

How do I prove $$\lim_{n\to\infty}(\sqrt[3]{a_n+1}-\sqrt[3]{a_n})=0 $$ where $a_n\to \infty$ using the standard definition of the limit, or in other words using $\epsilon$?
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1answer
25 views

Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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0answers
20 views

When is it possible to bound a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with $\big|\ f(x_1,x_2,\ldots,x_n)\ \big| \le {\prod}_{i=1}^n h_i(x_i)$

Is there any result that specifies when a multivariate function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ can be bounded (either locally or globally) by a product of some functions $h_i:\mathbb{R} ...
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0answers
6 views

What is the local form of the function at the point of self-intersection with a contour?

I am trying to solve this question : One of the contours (i.e. loci of locations with the same value) of a generic smooth scalar function of the two-dimensional plane is roughly ...
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1answer
26 views

Time derivative of logistic function [on hold]

I was wondering whether there is a possible solution to this. If we have function $$ y_t = \frac{x_t}{1+x_t}. $$ given that $x_t>0$ we can represent it as a logistic function $$ y_t = ...
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0answers
37 views

cn you help me with an answer to this question [on hold]

4\surd !\,27 - 8\surd !\,12 + \surd !\,3
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0answers
16 views

Combinatorics Question for generating fuctions [on hold]

Any tips/helps would be greatly appreciated! Let h_n be a number sequence where h_n = 3h_(n-1) - 2h_(n-2) with h_0 = 0 and h_1 = 1. Compute the ordinary generating function of h_n, and then compute a ...
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1answer
29 views

Combinatorics Generating Functions [on hold]

Any tips/comments would be greatly appreciated! Compute the generating function of the number sequence $h_n = (-2)^n n^2$ where $n\geq 0$.
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2answers
28 views

Verify that $\alpha(a)\neq2$ for all $a$ where $\alpha(x): (2x + 1)/(x + 2)$

If $A= \mathbb{R} \setminus \{-2\}$ and $B = \mathbb{R} \setminus \{2\}$, let $\alpha: A \to B$ by $\alpha(x): (2x + 1)/(x + 2)$. Verify that $\alpha(a)\neq2$ for all $a \in A$. As a hint, I was ...
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1answer
23 views

Linear operators in the polynomial function space

Let's say I have the standard monomial basis for P2: {$1$, $x$, $x^2$} And I have a linear operator $T: P_{2} \rightarrow P_{2}$ defined by: $T(f)(x) = f'(x) + xf''(x) + x^2f''(x)$ Now I want to ...
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1answer
13 views

The set of values for $p \in \mathbb{R}$ for which the function$f(x)=\sqrt{\log_7(\frac{2x^2+px+5}{x^2+1})}$ is defined for every real number is:

There are five possible answers. I don't know how to solve this, I can define for when $2x^2+px+5$ is bigger than $0$ but don't know how to proceed $A:(-4,4)$ $B: (-\infty,-4)$ $C: (-\infty,4)$ $D: ...
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1answer
22 views

finite dimensional vector spaces of functions left invariant by translation

Let $E$ be a finite dimensional vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall f \in E, \forall t \in \mathbb{R}, x \mapsto f(x-t) \in E$. Example of such spaces ...
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0answers
27 views

Every primitive of an odd function is even (proof)

I'd like to prove that every primitive of an odd function is even. This is my reasoning; FACT: 1: if f(x) is even, then f'(x) is odd [easy to prove]; 2: if f(x) is odd, then f'(x) is even ...
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1answer
32 views

Meaning of $t \mapsto \phi_t(x)$

The context may well be of assistance: Consider a differential equation $x'=f(x)$. Assume that $f:\mathbb R^n\to\mathbb R^n$ is continuously differentiable. Denote by ...
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40 views

PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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1answer
59 views

Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x))\ldots$ [on hold]

Problem : Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x)), \ldots$ and generally , ...
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0answers
18 views

Probability Density Function with Cumulative Distribution & Standard Deviation

Suppose that $X$ is a continuous random variable with probability density function given by $$f(x) = \begin{cases} a & 1 ≤ x ≤ 3 \\ 0 & \mbox{otherwise} \end{cases} $$ Then: (a) How do you ...
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1answer
53 views

Show that $f$ is one-to-one if and only if it is onto.

Suppose that $f$ is a function from A to B, where A and B are finite sets with $| A |= |B|$. Show that $f$ is one-to-one if and only if it is onto. How should I begin?
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0answers
27 views

Programming language for function analysis

what would you suggest as a programming language for working with functions and graphs. I need them to work with complex piecewise functions. P.S it is okay, if you suggest any software.
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1answer
25 views

X divided by Y, N times until a boundary is reached

not sure how to ask this but here is an example: X = 31.0 Y = 2.0 Z = 5.0 i want to keep dividing X by Y and the result of that again by Y and so on until i reach Z i will stop. assuming N is the ...
0
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0answers
18 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...