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

learn more… | top users | synonyms (1)

10
votes
4answers
1k views

Proof of linear independence of $e^{at}$

Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$, form a linearly independent set ...
10
votes
4answers
200 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
10
votes
3answers
133 views

How can you factor $x^4-6x^3+8x^2+2x-1?$

The original question is: Solve this equation for x: $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ I expanded and simplified it to get $$x^4-6x^3+8x^2+2x-1=0$$ Since neither -1 nor 1 are factors, it appears ...
10
votes
3answers
149 views

Regularity of the function $|x|^ax$

Assuming $x \in \mathbb{R}$, what can we say about the regularity class ($C, C^1, C^2, ..., \text{or}\ C^\infty$) of the following function (also with respect to $a \in \mathbb{R}$)? $$f(x)=|x|^ax$$
10
votes
2answers
365 views

Functions satisfying $f(m+f(n)) = f(m) + n$

I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you ...
10
votes
2answers
1k 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 \...
10
votes
1answer
969 views

What does it mean to extend a function?

What does it mean to extend a function? Can someone please give an example? Thanks in advance!
10
votes
2answers
131 views

Is a function of $\mathbb N$ known producing only prime numbers?

It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$. What about the function $f(n)=a^n+b$ ? Do positive integers $a$ and $b$ exists such ...
10
votes
1answer
12k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
10
votes
4answers
15k views

Find formula from values

Is there any "algorithm" or steps to follow to get a formula from a table of values. Example: Using this values: ...
10
votes
1answer
149 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
10
votes
2answers
1k views

Finite groups of functions under function composition

Over the years I have done many questions along the lines of the following: "Given functions $\phi, \theta$ (usually defined on $\mathbb{R}$ or $\mathbb{C}$, or a suitable subset of $\mathbb{R}$ or $\...
10
votes
1answer
1k views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
10
votes
1answer
1k views

Show f is uniformly continuous on $(a,b)$ if it is continuous and $\lim\limits_{x\to a^+}f(x)$ and $\lim\limits_{x\to b^-}f(x)$ exist

Let $f:(a,b)\to\mathbb{R}$ be continuous at all $x\in(a,b)$. If $\lim\limits_{x\to b^-}f(x)$ and $\lim\limits_{x\to a^+}f(x)$ exist in $\mathbb R$, how can we prove that $f$ is uniformly continuous on ...
10
votes
2answers
385 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and $g(x)$....
10
votes
1answer
464 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function (not necessarily continuous). Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. 1. Let $k\in\mathbb{...
10
votes
3answers
1k views

Infinitely differentiable function with given zero set?

For each closed set $A\subseteq\mathbb{R}$, is it possible to construct a real continuous function $f$ such that the zero set, $f^{-1}(0)$, of $f$ is precisely $A$, and $f$ is infinitely ...
10
votes
1answer
148 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...
10
votes
2answers
1k 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 ...
10
votes
2answers
311 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( \biggl\lfloor{\frac{2i^{2}}{p}\...
10
votes
1answer
173 views

If $f:X \to X$ is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous?

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a ...
10
votes
2answers
367 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.
10
votes
2answers
149 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 $...
10
votes
2answers
159 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ f_a(x)=f_a(...
10
votes
1answer
126 views

Simple true/false statement about function composition

Given the functions $f,g$ from $\mathbb{R}$ to $\mathbb{R}$ is it true that If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is monotonic I believe this is false but I can't ...
10
votes
1answer
272 views

Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$

Problem: Find all continuous real-valued functions $f$ such that $$f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}.\tag{1}$$ Here $f$ is allowed to be defined only on a subset of $\mathbb{R}$. The only ...
10
votes
1answer
233 views

Exercise about continuous functions

Consider a continuous function $f \, : \, [0,1] \, \longrightarrow \, [0,+\infty)$ such that $f(0)=f(1)=0$ and : $\forall x \in (0,1), \; f(x) > 0$. I would like to prove that there exist $x_{1},x_{...
10
votes
2answers
151 views

Example of function with a hundred minima

Find a function $f\colon\mathbb{R}^2 \to\mathbb{R}$, $f \in C^{\infty}$, such that $\nabla f = 0$ for exactly $100$ points and in these points there are only local minima.
10
votes
0answers
391 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 ...
10
votes
1answer
287 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
9
votes
8answers
2k views

“Integer average” of two integer numbers

Suppose two arbitrary integer numbers $a$ and $b$. I'm looking for some function $f(a,b)$ with the following properties: $f(a,b)\in\mathbb{Z}$. $f(a,a)=a$. $f(a,b)=f(b,a)$. $\min\{a,b\}< f(a,b)&...
9
votes
5answers
1k views

Is it possible to combine two integers in such a way that you can always take them apart later?

Given two integers $n$ and $m$ (assuming for both $0 < n < 1000000$) is there some function $f$ so that if I know $f(n, m) = x$ I can determine $n$ and $m$, given $x$? Order is important, so $f(...
9
votes
5answers
695 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
9
votes
4answers
839 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
9
votes
3answers
880 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
9
votes
5answers
2k 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 ...
9
votes
5answers
1k views

Is every injective function invertible?

Is every injective function invertible? How could I prove such thing? (Or is it just a necessary but not sufficient condition?) If $f:A\rightarrow B$ is injective then $f(a) = f(b) \Rightarrow a = b$ ...
9
votes
3answers
332 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
9
votes
5answers
834 views

Why do we limit the definition of a function? [duplicate]

Why do we limit the definition of a function to only one y per x? For example, the square root function. We only allow the principal square root of a number, rather than, say, the square root of 9 ...
9
votes
2answers
1k views

Draw graph of $\frac{1}{f(x)}$ from graph of $f(x)$

If I know the graph of $f(x)$, how do I draw the graph of $\frac{1}{f(x)}$?
9
votes
4answers
166 views

Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$

Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions? $$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
9
votes
3answers
613 views

For how many functions $f$ is $f(x)^{2}=x^{2}$?

How many functions $f$ are there that satisfy $f(x)^{2}=x^{2}$ for all $x$? My text (Spivak's Calculus; chapter 7 problem 7) asks this question for continuous $f$, for which the answer is, of course ...
9
votes
3answers
387 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?
9
votes
2answers
452 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 $x \in [0,1]$. I'll ...
9
votes
2answers
7k views

Example of a function continuous at only one point. [duplicate]

Possible Duplicate: Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point? I want to know some example of a continuous function which is continuous at ...
9
votes
6answers
204 views

Prove $\lfloor\frac{n+1}{2}\rfloor+\lfloor\frac{n+2}{4}\rfloor+\lfloor\frac{n+4}{8}\rfloor+\lfloor\frac{n+8}{16}\rfloor+ \dots=n$

Prove $$\left[\dfrac{n+1}{2}\right]+\left[\dfrac{n+2}{4}\right]+\left[\dfrac{n+4}{8}\right]+\left[\dfrac{n+8}{16}\right] + \dots=n$$ where $[x]=\lfloor x\rfloor$ $$$$ It was suggested that ...
9
votes
4answers
263 views

injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto (m+n)^{\max\{...
9
votes
4answers
1k 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 ...
9
votes
4answers
51k views

How to prove if a function is bijective?

I am having problems being able to formally demonstrate when a function is bijective (and therefore, surjective and injective). Here's an example: How do I prove that $g(x)$ is bijective? \begin{...
9
votes
2answers
520 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...