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

learn more… | top users | synonyms (1)

7
votes
1answer
357 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
3answers
110 views

How can we prove this function must be linear?

Let $f:[a,b]\to\mathbb{R}$ be continuous. Suppose for any sequence $(r_n)_{n=0}^{\infty}$ with $\lim_{n\to\infty}r_n=0$, and any $x\in(a,b)$: ...
7
votes
2answers
104 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
2answers
502 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
145 views

Determine an explicit expression for $f$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, bounded such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional. Determine an explicit ...
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
100 views

How to find all functions $f$ such that $f(a)+f(b)$ is square number, if $a+b$ is square number.

Question: For any $a,b\in N^{+}$, if $a+b$ is square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ satisfies the ...
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 ...
7
votes
0answers
278 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
203 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
747 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
1k views

how to see the logarithm as the inverse function of the exponential?

I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ...
6
votes
10answers
422 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
529 views

Why is there no function with a nonempty domain and an empty range?

Let $A$ to be a nonempty set and $B= \emptyset$; then $ A \times B$ is a set. And let $F$ be a function $A \to B$. Then $F \subseteq A \times B$. By the axiom of specification, $F$ must exists (if I ...
6
votes
5answers
477 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
440 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
270 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
654 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
298 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
361 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
215 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
513 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
413 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
612 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
625 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 ...
6
votes
3answers
1k 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
2answers
73 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
6
votes
2answers
107 views

Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
6
votes
3answers
148 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
257 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 $x \in [0,1]$. ...
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 ...
6
votes
3answers
256 views

Integral of a function defined in the set of Surreal Numbers

Given ${\{C}\}\ $ the set of all the $Surreal\ numbers$, is it possible to define the integral: $$\int_a^b{dxf(x)}$$where $$a\in{\{C}\},b\in{\{C}\},x\in{\{C}\}$$ Thanks
6
votes
3answers
551 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 ...
6
votes
3answers
2k 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
4answers
169 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
5answers
130 views

How prove $f(x)$ is a monotonic function if $f(x+y)=f(x)f(y)$

Let $f(x)$ be a real valued, differentiable function such that for any $x,y \in \mathbb{R}$,$f(x+y)=f(x)f(y)$. Suppose there exist $a,b$ such that $f(a)\neq 0, f'(b)>0$. Show that $f(x)$ ...
6
votes
4answers
123 views

The limit of $f$ or the limit of $f(x)$?

I have read before that $f$ denotes the function $f$ whilst $f(x)$ denotes the value of the function $f$ at $x$. What is right? To say that the limit of $f$ as $x$ tends to $a$ is $L$ or to say that ...
6
votes
1answer
265 views

Arithmetic function to return lowest in-parameter

Is there a mathematical function such that; f(3, 5) = 3 f(10, 2) = 2 f(14, 15) = 14 f(9, 9) = 9 It would be even more cool if there's a function that takes ...
6
votes
3answers
66 views

Proving $K$ is a group

Now I have proved certain things are a group before, and I know that it requires: 1)Associativity 2)Inverse 3)Identity But here I have such a strange thing that I wanted to clarify that I am doing ...
6
votes
6answers
201 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 ...
6
votes
3answers
179 views

Show that $\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$

Consider a function $f$ on non-negative integer such that $f(0)=1,f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \geq 2$. Show that $$\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$$ ...
6
votes
2answers
565 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 ...
6
votes
2answers
149 views

Does inverse of a nontrivial holomorphic function always have a branch point?

Any nontrivial (i.e. which is not a first order polynomial) entire in $\mathbb{C}$ function I have thought of has a multifunction as its inverse and has a branch point. For example, ...
6
votes
1answer
319 views

Understanding Hom functions

I am very new to category theory. Started learning about this Hom sets/functions. I read $\operatorname{Hom}(S,T)$ as set of all functions from $S$ to $T$ but somehow this is a overloaded definition ...
6
votes
4answers
137 views

Solution of $x^2 + s(x)\cdot x - n = 0$, with $s(x)$ is the sum of digits of $x$.

This problem comes from an programming competition website, but I'd interested in analyze it from mathematics prespective. Given this problem below, we must create a program that could give us the ...
6
votes
2answers
296 views

Is the “limit function” a continuous function?

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all $x_0\in\mathbb{R}$ we have $\lim\limits_{x\to x_0}f(x)=g(x_0)\in \mathbb{R}$. Is $g$ a continuous function?
6
votes
1answer
263 views

For a function from $\mathbb{R}$ to itself whose graph is connected in $\mathbb{R} \times \mathbb{R}$, yet is not continuous

In order to give an example of a function from $\mathbb{R}$ to itself whose graph is connected in $\mathbb{R} \times \mathbb{R}$, yet is not continuous, the book Berkley Problems on Mathematics refers ...
6
votes
2answers
116 views

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$ I've tried subbing in heaps of values but I keep getting things like ...
6
votes
2answers
2k views

Bijection from (0,1) to [0,1)

I'm trying to solve the following question: Let $f:(0,1)\to [0,1)$ and $g:[0,1)\to (0,1)$ be maps defined as $f(x)=x$ and $g(x)=\frac{x+1}{2}$. Use these maps to build a bijection ...
6
votes
2answers
145 views

What's the name for a bijection where pairs of elements map to each other?

First, a bit of context. About a quarter of an hour ago I came across one of those "Internet math puzzles" on Facebook that stated: If 1 = 5, 2 = 10, 3 = 15, and 4 = 20, then 5 = ? The answer was ...