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

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7
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2answers
200 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
7
votes
6answers
345 views

Rigorous Definition of “Function of”

When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a ...
7
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4answers
316 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 ...
7
votes
2answers
602 views

Existence of $\vee$ or $\wedge$ for non-monotonic functions

This question is inspired by a discussion in chat with wj32. We allow for equality in the definition of increasing and decreasing and call a function monotonic if it is increasing or decreasing. If ...
7
votes
3answers
235 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
7
votes
4answers
134 views

Regarding the notation $f: a \mapsto b$

While I have come to understand that $f:a\mapsto b$ means that for input from the set $a$, the function will return a value from the set $b$, I am curious as to how far one may "drag" this notation. ...
7
votes
4answers
156 views

Mustn't a function map every element of its domain to range (but not codomain)? [Richard Hammack, P228]

How to Prove It, D Velleman P226, P228: Suppose $f$ is a relation from $A$ to $B.$ Then $f$ is a function from $A$ to $B$ means: $\forall \; \color{#009900}{a \in A}, \exists \; ! \; b \in ...
7
votes
3answers
238 views

Finite at every point but unbounded on every interval

Is is possible that a function $f$ is finite at every point but unbounded on every interval? What if f is measurable?
7
votes
1answer
145 views

$f(x)^2 ≥ f(x + y)(f(x) + y)$ for no $f$?

Prove that there is no function $f : \mathbb{R}^+ → \mathbb{R}^+$ such that $$f(x)^2 ≥ f(x + y)(f(x) + y)$$ for all $x, y > 0$. I can't think of a way of solving this.
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votes
3answers
177 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?
7
votes
2answers
211 views

Prove that between two roots of $f(x)$ there is a root of $g(x)$

Let $f(x),g(x)$ be differential functions, and $f'(x)g(x)\neq f(x)g'(x)$ for all $x\in\mathbb R$. Prove that between two roots of $f(x)$ there is a root of $g(x)$. I guess this has to do with Rolle's ...
7
votes
3answers
239 views

Category theory without codomains?

A surjection is a function whose range equals its codomain. Thus, the distinction between functions and surjections requires the notion of a codomain. Similarly, a bijection is an injection whose ...
7
votes
2answers
447 views

Need mathematical function for “adding” 0.5 and 0.5 and getting 0.4

I'm looking for a mathematical function that would have the following attributes: Reasonably smooth -- continuous to the second or third derivative, say, for values greater than zero. Given two ...
7
votes
4answers
233 views

Why does $\ln(x) = \epsilon x$ have 2 solutions?

I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ ...
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votes
3answers
1k 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 ...
7
votes
1answer
443 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
7
votes
3answers
254 views

Show that $\frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )}$ is decreasing in $y > 1 $.

I am interested in the function $f(y) = \frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )},$ for values of $c \in (0,1)$, and $y > 1$, and have been trying to show that the function is decreasing. I ...
7
votes
2answers
99 views

How to find period of $f$ if $f(x+13) + f(x+630) = 0$

Let $f:\Bbb{R}\to\Bbb{R}$ be a periodic function with period $T$. The question was to find the (fundamental) period given the following relation. $$ f(x+13) + f(x+630) = 0 $$ Now, the given method ...
7
votes
2answers
238 views

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities?

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities? In functions like $\tan^{-1}(a \tan(x))$, the angle wraps and the result is discontinuous. Is it legitimate to redefine the ...
7
votes
1answer
313 views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
7
votes
4answers
561 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
7
votes
1answer
327 views

Find $f(5)$ of a non-constant polynomial function $f(x)$

Suppose $f(x)$ is a non-constant polynomial such that $f(x^ 3) − f(x ^ 3 − 2) = f( x )\cdot f(x) + 12$ for all $x$. Find $f(5)$? I find this problem on Quora just now, and I try to solve it but do ...
7
votes
1answer
199 views

Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$

Let $(a,b) \in \mathbb{R}^{2}$ such that $a<b$ and $f\in C^2([a,b],\mathbb{R})$ such that $f'\neq 0$ and $f''/f'$ is decreasing. Prove that : $$|f(b)-f(a)|\geqslant (b-a) ...
7
votes
2answers
475 views

Proving Injectivity

The problem is to show the function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by $$f(x,y)=(\tfrac{1}{2}x^2+y^2+2y,\,x^2-2x+y^3)$$ is injective on the set ...
7
votes
1answer
196 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
7
votes
2answers
97 views

an injection into $\mathbb{N}$

Is that true that the map $f\colon \{(m,n)\in\mathbb N^2:m\le n\}\to\mathbb N$ defined by $(m,n)\mapsto (m+n)^{\max\{m,n\}}$ is an injection? If it is, how to prove that? I have asked a similar ...
7
votes
1answer
1k views

Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.

Given any set $X$, let $P(X)$ be the set of all subsets of $X$, and let $\{0,1\}^X$ be the set of all functions $X \rightarrow \{0,1\}$. Construct a bijection (and its inverse) between P(X) and ...
7
votes
1answer
199 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 ...
7
votes
0answers
253 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
7
votes
0answers
200 views

When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
6
votes
5answers
719 views

Do we have always $f(A \cap B) = f(A) \cap f(B)$?

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$? Thanks in advance
6
votes
10answers
409 views

Sanity check, is $\{(-9,-3),(2,-1),(7,7),(-1,-1)\}$ a function?

EDIT#2: Yes, I'm crazy! This IS a function. Thanks for beating the correct logic into me everyone! I'm using a website provided by my algebra textbook that has questions and answers. It has the ...
6
votes
5answers
461 views

Does $\sin(t)$ have the same frequency as $\sin(\sin(t))$?

I plotted $\sin(t)$ and below it $\sin(\sin(t))$ on my computer and it looks as if they have the same frequency. That led me to wonder about the following statement: $\sin(t)$ has the same ...
6
votes
4answers
394 views

Is a Bijection From a Group to Itself Automatically an Isomorphism If It Maps the Identity to Itself?

I am looking at $\operatorname{Aut}(V)$, where $V$ is the Klein 4-group. I noticed that $\operatorname{Aut}(V)$ is comprised of all the permutations of the elements of $V$ where $1$ is mapped to ...
6
votes
3answers
251 views

bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate]

I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I ...
6
votes
2answers
592 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 ...
6
votes
4answers
293 views

Is there a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$?

For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{Q}_{\geq 0}$?
6
votes
4answers
352 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 $g(g(g(x)))\equiv g^{3}(x)=x$? ...
6
votes
2answers
214 views

Is There A Function Of Constant Area?

If I take a point $(x,y)$ and multiply the coordinates $x\times y$ to find the area $(A)$ defined by the rectangle formed with the axes, then is there a function $f(x)$ so that $xy = A$, regardless of ...
6
votes
3answers
459 views

Are injectivity and surjectivity dual?

Are injectivity and surjectivity dual in some sense? Their set-theoretic definitions are quite different. In particular, the injectivity is a property of a function's graph, while surjectivity is a ...
6
votes
2answers
396 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
6
votes
4answers
608 views

$f: \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2)=5$

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2) = 5$.
6
votes
2answers
3k views

Constructing Continuous functions at given points

Ok. This question may sound very easy, but actually i am in great need of it. I have been facing trouble in constructing functions, which are only continuous at some particular sets. For e.g, the ...
6
votes
3answers
146 views

Prove that $f(x)=x$ if the following holds true [closed]

Let $f\colon\mathbb R \to \mathbb R$ be a continuous odd function such that 1) $f(1+x)=1+f(x)$ 2) $x^2f(1/x)=f(x)$ for $x\ne0$. Prove that $f(x)=x$.
6
votes
2answers
113 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 ...
6
votes
3answers
474 views

What is the set-theoretic definition of a function?

I'm reading through Asaf Karagila's answer to the question What is the Axiom of Choice and Axiom of Determinacy, and while reading the explanation of Bertrand Russell's analogy ("The Axiom of Choice ...
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3answers
1k views

Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

In matters of conic sections, are there other properties such that it helps to group the circle and hyperbola in one, and the parabola and ellipse in the other?
6
votes
3answers
957 views

Lagrange Multipliers

I would like to find the extrema of the function $f(x,y)=x^2+4xy+4y^2$ subject to $x^2+2y^2=4$ using Lagrange Multipliers. Is it possible to get for the Lagrange multipliers the value zero? I don't ...
6
votes
4answers
167 views

Function behavior with very large variables

Whenever I think about how a function behaves, I always try to identify a general pattern of behavior with some common numbers (somewhere between 5 and 100 maybe) and then I try to see if anything ...
6
votes
2answers
236 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 $x \in [0,1]$. ...