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

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3
votes
1answer
86 views

What are the odds a graph generated “randomly” represents a function?

I'm creating an activity for my class that teaches the vertical and horizontal line tests and for the life of me I can't figure out the probabilities involved. Here's the exercise: Draw 5 pairs of ...
6
votes
2answers
117 views

Continuation of strictly monotone functions on $\mathbb{R}$

While studying the properties of ordinal utility functions, I came across the following question. Given a strictly increasing function $f : D \rightarrow \mathbb{R}$, where $D$ is an arbitrary ...
0
votes
2answers
275 views

Functions that literally flatten out

Is there a way to write a function so that for any $c$ such that $a < c < b$, $f(c)$ is always the same? For example, if you had an increasing function up until $0$ at which point the $f(x)$ ...
6
votes
4answers
362 views

Can the Identity Map be a repeated composition one other function?

Consider the mapping $f:x\to\frac{1}{x}, (x\ne0)$. It is trivial to see that $f(f(x))=x$. My question is whether or not there exists a continuous map $g$ such that $g(g(g(x)))\equiv g^{3}(x)=x$? ...
2
votes
2answers
135 views

Subset of a function

Suppose we have a function $f:X \rightarrow Y$. Now, consider the function $g:X'\rightarrow Y$ where $X'\subset X$. I'd like to say the $g$ is a "subset" of $f$ ; is there a correct term for ...
4
votes
4answers
300 views

Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
1
vote
2answers
150 views

Is using a step function to limit the value of a function considered inelegant?

I am a programmer and recently we had some problems with a certain function $f$ that tends to infinity near 10, and we can not have such great values as a result. Recently there was a collegue trying ...
1
vote
1answer
384 views

Boundedness of continuous functions depending on whether an interval is open or close

Firstly, I haven't done any math in 4 months so I am bit rusty. Is there a difference between the boundedness of a continuous function depending on whether we are taking it over a closed or open ...
1
vote
1answer
101 views

Function $f(x,y)=u(x)v(y)$ differentiable of 2 variables

If $u(x)$ is real and differentiable and $v(y)$ is real and differentiable, is it then given that $f(x,y) = u(x)\cdot v(y)$ is differentiable? I have tried to make a proof in the case where ...
1
vote
1answer
36 views

How do you call a convention by which binary operator is an unary one returning another unary operator?

How do you call a convention by which any binary operator is an unary one returning another unary operator? $F(x,y)\equiv (F(x))(y)$ This convention is used in functional programming languages. I ...
0
votes
1answer
66 views

Joe the incremental car pusher

Whenever I call my friend Joe, he pushes a car down a road. (Assume the road is flat and has no other cars on it, and that the car has no brakes.) The car Joe's pushing has to cover a very precise ...
4
votes
1answer
98 views

Problem with uniform convergence

Let $f_n$ ($n=1,2,\dots$) be a sequence of functions $f_n\colon \mathbb R\to \mathbb R$ of class $C^1$ such that $f_n \rightrightarrows 0 $, $f_n' \rightrightarrows 0 $. Assume moreover that functions ...
1
vote
5answers
1k views

Example of a bijection between two sets

I am trying to come up with a bijective function $f$ between the set : $\left \{ 2\alpha -1:\alpha \in \mathbb{N} \right \}$ and the set $\left \{ \beta\in \mathbb{N} :\beta\geq 9 \right \}$, but I ...
-3
votes
9answers
278 views

Why is $y + x = 3$ not the same as $y^2 + x^2 = 9$

I know this is impossible, but why is the following not possible: $y + x = 3$ is the same as $y^2 + x^2 = 9$ They're meant to be equivalent.
3
votes
1answer
446 views

If $F$ and $G$ are one to one, then $G \circ F$ is one to one and $(G \circ F)^\neg = F^\neg \circ G^\neg$

THEOREM: if $F$ and $G$ are one to one then $G \circ F$ is also one to one and $(G \circ F)^\neg$ = $F^\neg \circ G^\neg$ PROOF: if $F: A\rightarrow B$, $G: B \rightarrow C$ and $$\forall a, a' ...
7
votes
1answer
203 views

A question about showing $f(x)=0$

Let $f$ be a function from the set of real numbers to itself that satisfies $f(x + y) ≤ yf(x) + f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x ≤ 0$. I tried to show that ...
1
vote
0answers
131 views

Properties of the sum or product of functions

I have $m$ functions of $n$ variables, $f_i(x_1,\dots,x_n)$ (where both $m$ and $n$ are finite), and I want to find the maximum (or the minimum) of: \begin{equation} G=\sum_{i=1}^{m}f_i(\bf{x}) ...
3
votes
0answers
104 views

On which of the following spaces is every continuous (real-valued) function bounded

On which of the following spaces is every continuous (real-valued) function bounded? i) $X_1 = (0, 1)$; ii) $X_2 = [0,1]$; iii) $X_3 = [0, 1)$; iv) $X_4 =\{t \in [0, 1] : t \mbox{ ...
4
votes
1answer
80 views

If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.

THEOREM: If $ \text{ran } F \subseteq \text{dom } G $ then $\text{dom }(G \circ F)= \text{dom }F$ PROOF: if $ F\subseteq A \times B$ and $ G\subseteq B\times C$ then by definition ...
3
votes
1answer
390 views

How to find a summation of a logarithmic function?

Suppose that I had to find $\log_{10}(8952!)$. Now, since $\log(a) + \log(b) = \log(ab)$, this can be rewritten to the following summation: $$\sum_{x=1}^{8952}{\log_{10}(x)}$$ Would there be a ...
1
vote
2answers
303 views

To prove mapping f is injective and the other f is bijective

There's a mapping $f:X\rightarrow Y$. 1.for all $A,B\subset X$, $f(A\cap B)=f(A)\cap f(B)$, prove $f$ is injective. 2.for all $A\subset X$, $f(A^{c})=[f(A)]^{c}$, prove $f$ is bijective.
1
vote
1answer
51 views

Trouble with function transformation (Left and right)

I am reading this example in the book for Pre-Calculus and it is explaining how functions are shifted left or right using g(x)=f(x-1). Here is what it says in the book. Define a function g by g(x) = ...
0
votes
1answer
187 views

Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following: $\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous. ...
4
votes
3answers
213 views

Help solving this question on even and odd functions

a) Suppose that $E(x)$ is an even function and that $O(x)$ is an odd function. Suppose furthermore that $E(x) + O(x) = 0$. Show that for all $x$, $E(x) = 0$ and $O(x) = 0$. b) Use part a) to show ...
0
votes
3answers
155 views

Induction on functions

I'm working through some homework on induction, and most problems I can solve fine, but I have problem getting started on induction proofs that ask you to prove function relations. For example, here ...
5
votes
2answers
119 views

Function which is $2^n$ - periodic for all integers $n$

Is there a non-constant, continuous function $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that for all integers $n$, $f$ is $2^n$-periodic? Notes: $n$ can be any integer, and so can be negative ...
3
votes
1answer
305 views

All asymptotes of $f(x)=\sqrt{x^2-4x}+\frac{1}{x^2-1}$

Find the number of all possible asymptotes of: $$f(x)=\sqrt{x^2-4x}+\frac{1}{x^2-1}$$ Since we know $\sqrt{ax^2+bx+c}\approxeq \sqrt{a}\big|x+\frac{b}{2a}\big|$ when $(x\rightarrow\pm\infty)$ so, ...
5
votes
2answers
343 views

Are all multiplicative functions additive?

Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$. Proof: There is some $c$ such that $y=cx$. Then ...
2
votes
5answers
390 views

Converting an IF condition to a mathematical equation

I am trying to study about converting algorithms into mathematical equations. For this I just started with a simple random example : ...
0
votes
2answers
325 views

Integrating Out

In probability integrating out a variable is viewed as marginalization; One probability function turns into another probability function. In other cases and fields, taking a regular function as ...
0
votes
1answer
99 views

Linear independence of $\{\exp(b_{n}z):n\in\mathbb N\}$

I have the following question: Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for some complex point $z\in\mathbb Z$. Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ ...
8
votes
1answer
269 views

Function $f:\mathbb R^+\rightarrow \mathbb R^+$ that is eventually greater than $x^{x^{x^{…^{x^x}}}}$

For each $n$, define $f_n:\mathbb R^+\rightarrow \mathbb R^+$ by $f_n(x) = \underbrace{x^{x^{x^{...^{x^x}}}}}_n$ I want to find a function $f:\mathbb R^+\rightarrow \mathbb R^+$ such that for any ...
2
votes
2answers
154 views

True statements for a continuous function

Let $f\colon \mathbb R\rightarrow \mathbb R$ be a continuous function. Define $G = \{(x, f(x)) : x \in \mathbb R\} \subseteq \mathbb R^2$. Pick out the true statements: a. $G$ is closed in $\mathbb ...
0
votes
1answer
2k views

Calculation of bessel function versus matlab solution

I am looking to calculate the Bessel function of the first kind $J_o(\beta)$. I am using the formula (referenced from wikipedia) to accomplish this. $$J_\alpha (\beta) = ...
3
votes
1answer
189 views

Minimizing a function over two variables

Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let $$ \psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} - ...
0
votes
1answer
308 views

Find function of given 3 inputs and output min, max

i1 16 64 400 25 8 8 i2 1 1 1 4 3 3 i3 4 20 40 4 100 200 min 10 15 35 14 30 35 max 10 20 45 16 30 45 How can I get the function ...
3
votes
1answer
91 views

Find whether or not an inverse exists algebraically

Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus? I'm an undergrad engineer and can obviously solve this using basic calculus, but ...
2
votes
0answers
217 views

Implication of Lipschitz continuity

I am a little bit unsure about the following claim. Let $H:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ be a mapping of the form $H(x,f(x))$, where $f:[0,1]\rightarrow \mathbb{R}$ is some continuous ...
1
vote
1answer
130 views

Set Theory in beginning topology

Here is the set-up. Suppose that $\left \{ A_\alpha : \alpha \in \Lambda \right \}$ is a collection of subsets of a set X and $\left \{ B_\beta : \beta \in \Psi \right \}$ is a collection of subsets ...
3
votes
2answers
694 views

Image set of a function.

I am revising some old exam question that i have now reflecting back on it.. the question is to find the image set of the function.. I forgot how to do this.. can some one tell me the steps not just ...
2
votes
1answer
227 views

Is there a geometric projection for every complex function

I was wondering about the best way to visualize complex functions. As they're $$ R^2 \rightarrow R^2$$ i think best way are complex plane image/grid transforms like they used in the Dimensions movie ...
4
votes
2answers
190 views

Triple “Pythagorean identity”

It's not hard to find multiple trigonometric functions of period $2\pi$ that added to self shifted by some constant offset result in a constant. In classic pythagorean identity, you have ...
12
votes
2answers
178 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
1
vote
2answers
194 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
1
vote
1answer
637 views

Prove that a set is Bounded

In the process of constructing a proof, I obtain the following set. Let C be the set of continuous and nondecreasing functions defined on $[0,1]$. Let the set $B$ be given as follows: $$ ...
2
votes
2answers
178 views

Place of functions in order of operations?

So I came upon something that I thought was interesting, and my google search didn't come up with a good answer. Say you have some function $f(x)$. Where does evaluating $f(x)$ come in the order of ...
0
votes
2answers
114 views

solving $\mathrm{e}^x x^n =1- c x$ for $x$

I'm struggling with solving $x$ for the following equation: it looks in the form as $$e^x x^n=1-cx$$ where $c$ is just some constant number. I tried to solve $x$ explicitly. But it seems not be able ...
-5
votes
3answers
2k views

Associative Property for Composition of Functions

Please explain me proof of "Associative Property for Composition of Functions"
0
votes
1answer
153 views

Naming conventions for super- and subscripts when naming sets and functions

Ok, so say I want to create a bunch of sets and functions for my to-be paper (that surely will get the attention of those comity members in Stockholm), and I want to identify them with the help of ...
1
vote
2answers
150 views

Random Sequence Generator function

I want to find out a function or algorithm, whichever is suitable, which can provide me a random sequence. Like Input: 3 Output: {1,2,3} or {1,3,2} or {2,1,3} or {2,3,1} or {3,1,3} or {3,2,1} Same ...