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

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9
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3answers
252 views

Quadratic Equation Recurrence?

So I was playing around with the following: Let $f_0(x) = x^2 - bx + c$. If $f_n(x)$ has roots $p$ and $q$ with $p > q$, then let $f_{n+1}(x) = x^2 - px + q$. The recurrence relation is rather ...
9
votes
1answer
102 views

$\max(a,b)=\frac{a+b+|a-b|}{2}$ generalization

I am aware of an occasionally handy identity: $$\max(a,b)=\frac{a+b+|a-b|}{2}$$ However, I have found I'm unable to come up with a nice similar form for $\max(a,b,c)$. Of course I could always use ...
9
votes
1answer
120 views

Non-computable function having computable values on a dense set of computable arguments

A rational complex number is a complex number whose both real and imaginary parts are rational numbers. Note that a rational complex number is a finitary object that can be an input or an output of an ...
9
votes
2answers
136 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
9
votes
1answer
86 views

Interesting properties of the function $(a,b)\mapsto a/(a-b)$

Consider the extremely simple function $$f(a,b)=\frac a{a-b}.$$ This gives the coordinate where the line through $(0,a)$ and $(1,b)$ meets the $x$-axis. I noticed that the function $f$ has some ...
9
votes
2answers
172 views

What is the name of this property of a function

I'm trying to find the right vocab word to describe a concept: In computational geometry, there's a concept of a polygon "monotone" with respect to a line. Which means that the polygon intersects ...
9
votes
2answers
244 views

How do I find, algorithmically, which parts of a given function are interesting to graph?

I'm building a program that does 2D graphing, and was wondering: How can I determine the default zoom level and x/y extents to display on screen, in such a way as to maximise the 'interesting' parts ...
9
votes
0answers
361 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
8
votes
8answers
276 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
8
votes
6answers
5k views

Max Min of function less than Min max of function

I can't understand why max min of a function is less than equal to min max of that function i.e Why $$\underset{x}{\text{max}}\:\underset{y}{\text{min}} f(x,y) \leq ...
8
votes
5answers
1k views

What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...
8
votes
6answers
1k views

what is this sort of function called?

I am doing an assignment but I do not know how to do this problem. I have the following: $$ f(x)= \begin{cases} 0 & \text{for $x<0$},\\ x & \text{for $x\geq 0$}. \end{cases} $$ We are ...
8
votes
8answers
581 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
8
votes
3answers
651 views

Why is there no continuous log function on $\mathbb{C}\setminus\{0\}$?

Over the years, I've often heard that there is no logarithm function which is continuous on $\mathbb{C}\setminus\{0\}$. The usual explanation is usually some handwavey argument about following such a ...
8
votes
4answers
153 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
8
votes
8answers
703 views

How to denote “powers” of a function?

I'm working with functions themselves, and I have learned that functional powers mean composition so: $f^3 = f \circ f \circ f$ But I'm looking for something that means $fff$. So $(fff)(x) = ...
8
votes
2answers
982 views

Why use the derivative and not the symmetric derivative?

The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
8
votes
4answers
544 views

Is $\tan\theta\cos\theta=\sin\theta$ an identity?

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is ...
8
votes
4answers
423 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 $g(g(g(x)))\equiv g^{3}(x)=x$? ...
8
votes
2answers
648 views

Is a function changed into another function by a change of variables?

If I have a function $ u(x,t) = p(x+ct) + q(x-ct) $ (which is the d'Alembert solution to the $1D$-wave equation), I can make the substitutions $$ \xi(x,t) = x + ct\\ \eta(x,t) = x - ct $$ So I am ...
8
votes
3answers
1k views

Proof of a simple property of real, constant functions.

I recently came across the following theorem: $$ \forall x_1, x_2 \in \mathbb{R},\textrm{function, } f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto y; \ |f(x_1) - f(x_2)| \leq (x_1-x_2)^2 \implies ...
8
votes
5answers
594 views

General Introduction to Functional and other Mathematic Notations

I've been a programmer for a good while now. Fairly experienced at a bit of math as far as coming up with algorithms and such but I am far far behind on understanding quite a deal of notation. Here ...
8
votes
5answers
595 views

Study continuity of this function

Hello im studying calculus at the university and I dont know how to solve the following exercise: Study the continuity of the next function: $$f(x,y) = \begin{cases} \frac{x^2-xy}{x+y}&\text{for } ...
8
votes
1answer
283 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 ...
8
votes
5answers
480 views

Does there exist a function $f:[0,1] \to[0,1]$ such its graph is dense in $[0,1]\times[0,1]$?

Does there exist a function $f:[0,1]\to [0,1]$ such that the graph of $f$ is dense in $[0,1]\times[0,1]$? Not necessarily continuous.
8
votes
4answers
297 views

Is it known or where does this lead to?

I am eleventh class student, recently I started learning calculus. I was experimenting on various things, and found a new thing. It is as follows. Let us consider a function $f(x)$which is ...
8
votes
2answers
4k views

What is a function to represent a diagonal sine wave?

I need to be able to plot pixels in this pattern. To me, it looks like a sine wave pattern that is both diagonal and convergent. What would a function for that look like? Thanks.
8
votes
2answers
417 views

Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$

Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that $|f'(x)| \le A/2$ for every $x \in [0,1]$. ...
8
votes
4answers
714 views

Is the following a valid mathematical statement?

For all $f:\mathbb N\to\{1,2,3,\ldots,100\}$, If $f$ is a one to one correspondence, Then $f^{-1}(2)=3$ It seems as though this should not be a valid statement, since the implication fails to ...
8
votes
4answers
772 views

Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$. Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$ Therefore, $y\in F(A)$ and ...
8
votes
3answers
347 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
8
votes
2answers
55k views

Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?

I would very much like to have a complete list of the types of polynomial functions. I know that theres: ...
8
votes
2answers
283 views

Odd $C^\infty$ function

Suppose that $$f\in C^\infty (\mathbb{R})$$ and $f$ is an odd function. ($f(x)=-f(-x)$) What can we say about the zero at zero? Does $f$ have to be of the form $x g(x)$ for some $g\in C^\infty ...
8
votes
3answers
291 views

$f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice

Does there exist a continuous function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ exactly two times?
8
votes
2answers
813 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
8
votes
4answers
278 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
8
votes
1answer
85 views

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$ $(A)\frac{1}{10}\hspace{1 cm}(B)10\hspace{1 cm}(C)1\hspace{1 cm}(D)$cant be obtained I could not solve this question.I ...
8
votes
2answers
164 views

What is the meaning of $\mathbb{R}\setminus\{0\}$?

This is used in many posts related to functions and googling it doesn't help. What does this mean? $\mathbb{R}$ should stand for all Real numbers.
8
votes
1answer
676 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
8
votes
7answers
241 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
8
votes
3answers
81 views

$(\Bbb R \to \Bbb R : x\mapsto x^2)\equiv(\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2) \not\equiv (\Bbb C \to \Bbb C:x\mapsto x^2)$

Consider the following functions: $f:\Bbb R \to \Bbb R : x\mapsto x^2$ $g:\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2$ $h:\Bbb C \to \Bbb C:x\mapsto x^2$ I'm quite sure that $h$ is not equal to $f$ ...
8
votes
3answers
198 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
8
votes
3answers
245 views

is the following a decreasing function?

I am stuck on figuring out why the following function is a decreasing function when I read a paper. The function is following $$f(x)=-\frac{1}{x}\log[{pe^{-ax}+(1-p)e^{-bx}}]$$ where $a$ and $b$ are ...
8
votes
3answers
2k views

chain rule using tree diagram, why does it work?

In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to ...
8
votes
2answers
190 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
8
votes
4answers
136 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: $2^x=8$ ...
8
votes
1answer
120 views

Find $f$ such as $f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$

Find $f \in C^0([0,1] , \mathbb{R})$ such as $$f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$$ My try : Constant functions work fine. We can notice : $$f(x) = \frac{f(x)}{2}+\sum_{n=2}^\infty ...
8
votes
2answers
422 views

if $|f(n+1)-f(n)|\leq 2001$, $|g(n+1)-g(n)|\leq 2001$, $|(fg)(n+1)-(fg)(n)|\leq 2001$ then $\min\{f(n),g(n)\}$ is bounded

The following question was proposed at MOP 2001 A function $f:\mathbb{N}\to\mathbb{N}$ is called cautious if $|f(n+1)-f(n)|\leq 2001$ for all $ n\in\mathbb{N} $. Suppose that $f,g,h$ are ...
8
votes
2answers
108 views

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$? Personally I would say: "no". In my view a function can only ...
8
votes
3answers
269 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...