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

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6
votes
4answers
118 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
252 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
2answers
96 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
177 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
113 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
4answers
135 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
291 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
258 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
1k 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
137 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 ...
6
votes
3answers
600 views

Is there a one-one onto continuous function $f:[0,1]\rightarrow[0,1]^2$?

Is there a one-one onto continuous function $f:[0,1]\rightarrow[0,1]^2$? I was trying to prove that there is no such function, but failed. Any suggestions?
6
votes
3answers
497 views

Describing a Wave

I have this wave in front of me, and I am to describe this into a math description such as its function that is equivalent to representing this wave. I have no idea how to start and could use some ...
6
votes
1answer
176 views

$f(a(x))=f(x)$ - functional equation

I was reading "Functional Equations and How to Solve Them" by Small and the following comment pops up without much justification on p. 13: If $a(x)$ is an involution, then $f(a(x))=f(x)$ has as ...
6
votes
3answers
210 views

Learning Math At Home

I want to learn math on my own at home. What is the best method to do so? I would say that I pick things up/grasp concepts pretty fast. I took math until grade 10 in highschool/secondary school and ...
6
votes
1answer
83 views

if $f\left(x+y,\frac{y}{x}\right)= x^2-y^2$ then $f(x,y)=?$

So, I have to find $f(x,y)$ if the following holds: $$f\left(x+y,\frac{y}{x}\right)= x^2-y^2$$ I thought about replacing $x+y=X$, and $y/x=Y$, but now where do I replace this $x$ and $y$ that I've ...
6
votes
3answers
936 views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
6
votes
1answer
152 views

Notation for functions vs. numbers

I understand that $f$ represents a function while $f(x)$ represents the value of a function, but while I can easily see how to apply this convention to work with functions instead of numbers in some ...
6
votes
2answers
99 views

Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
6
votes
2answers
92 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
6
votes
1answer
607 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 ...
6
votes
2answers
167 views

System of two Equations

A friend of Mine gave me a system of two equations and asked me to solve them $\rightarrow$ $$\sqrt{x}+y=11~~ ...1$$ $$\sqrt{y}+x=7~~ ...2$$ I tried to solve them manually and got this horrendously ...
6
votes
3answers
444 views

Teaching the concept of a function.

I am doing a class for at risk high school math students on the concept of a function. I have seen all the Internet lesson plans and different differentiated instruction plans. The idea of a ...
6
votes
3answers
159 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
6
votes
2answers
286 views

Function Olympiad Problem: Define $f(n)$ such that $f(n)$ is a positive integer, $f(n+1)$ $>$ $f(n)$ and $f(f(n))$ $=$ $3n$. The value of $f(10)$ is?

If you know me at all, or have read my profile, or have seen any of my previous questions, you might know that I am very interested in Olympiad maths and have come across many challenging maths ...
6
votes
4answers
426 views

Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$

This came out of a textbook problem, and as Lubin pointed out below, it's not actually true as originally stated. I'm guessing it should be restated as: If the graphs of $y=f(x)$ and $y=f^{-1}(x)$ ...
6
votes
2answers
321 views

equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
6
votes
2answers
155 views

$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous

I'm doing an exercise that asks me to prove that $f$ is continuous using a $\epsilon$-$\delta$ proof. I have that $$ f(x) = \begin{cases} x\cdot \sin \frac1x,&x\neq 0 \\ 0,&x = 0 \end{cases} ...
6
votes
2answers
249 views

Currious property of monotonic functions

If $f(x)$ is continuous and monontonicly increaseing on the interval $[1,\infty]$, and $f'(x)\leq\frac{1}{x}$ on the interval $[1,\infty]$, is $$\lim_{n \rightarrow \infty} ...
6
votes
1answer
207 views

When to create transcendental function to solve “unsolvable problem”?

$\int \frac{1}{x} dx$ is an unsolvable problem using standard laws of Calculus (power rule) without the use of the function $f(x) = \ln x$ which was handcrafted by mathematicians to solve such ...
6
votes
2answers
284 views

$f(n)=$ how many times number $2$ appears in numbers from $1$ to $n$

$f(n)$ calculates how many times $2$ appears in numbers from $1$ to $n$. For example, $f(1)=0$, $f(2)=1$ and $f(12)=2$. What is the first $n$ for which $f(n)=n$? I would need first of all to figure ...
6
votes
1answer
313 views

Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
6
votes
2answers
398 views

What is an algebraic function?

I am doing first year university calculus, and we are learning about the different kinds of functions. According to wikipedia, an algebraic function ...
6
votes
2answers
2k views

Change of order of double limit of function sequence

The more general quesion is under what conditions the folloing equality will hold (all functions are real valued): $$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow ...
6
votes
1answer
176 views

Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$

Let $f\in C^2(\mathbb{R},\mathbb{R})$. Assume there exist $(a,b) \in \mathbb{R}^2$ such that $\forall x \in \mathbb{R}, (f(x))^{2} \leqslant a$ and $(f'(x))^{2} + (f''(x))^{2} \leqslant b$. ...
6
votes
4answers
1k views

History of $f \circ g$

$f \circ g$ is usually interpreted as $f(g(x))$ although, as Google shows, $g(f(x))$ is used frequently too. My question: Does anybody know who was the first mathematician to use this symbol and what ...
6
votes
2answers
501 views

Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input

$\alpha _n ^n-1=0$ $\alpha _n=e^{2 \pi i/n}$ $$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$ I have read in Jim Brown's paper on page 5 that ...
6
votes
2answers
225 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
6
votes
2answers
193 views

Prove that the function $f(x,y) = ax + by$ is onto

I have been thinking about this problem for a while and have gotten stuck. This is a homework question so I just require some hints to push me to the answer. Question: Let $a, b$ be integers. ...
6
votes
1answer
76 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
6
votes
3answers
90 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 $x\in(a,b)$: ...
6
votes
3answers
259 views

Find the range of: $y=\sqrt{\sin(\log_e\frac{x^2+e}{x^2+1})}+\sqrt{\cos(\log_e\frac{x^2+e}{x^2+1})}$

Find the range of: $$y=\sqrt{\sin(\log_e\frac{x^2+e}{x^2+1})}+\sqrt{\cos(\log_e\frac{x^2+e}{x^2+1})}$$ What I tried: Let:$$\log_e\frac{x^2+e}{x^2+1}=X,$$ then $$y=\sqrt {\sin X}+\sqrt{\cos X}$$ ...
6
votes
2answers
209 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
6
votes
1answer
221 views

Exercise Functional Analysis

Let $\mathcal{F}$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider an operator $\mathcal{O}: \mathcal{F} \rightarrow \mathcal{F}$ such that: $\mathcal{O}( f_1 + f_2) = ...
6
votes
1answer
105 views

Prove that any function can be represented as a sum of two injections

How to prove that for any function $f: \mathbb{R} \rightarrow \mathbb{R}$ there exist two injections $g,h \in \mathbb{R}^{\mathbb{R}} \ : \ g+h=f$. Could you help me?
6
votes
1answer
325 views

Infinitely times differentiable function

Let $f$ belong to $ C^{\infty}[0,1]$ and for each $x \in [0,1]$ there exists $n \in \mathbb{N}$ so that $f^{(n)}(x)=0$. Prove that $f$ is a polynomial in $[0,1]$. I am trying to use Baire Category ...
6
votes
2answers
112 views

Representation of smooth function

Is it true that any smooth function $f\colon \mathbb{R}^n \to \mathbb{R}^n$ can be represented as $$ f(x) = \nabla U(x) + g(x) $$ where $U(x)$ is a scalar function and $\langle g(x), f(x) \rangle ...
6
votes
1answer
242 views

Complex Analysis: Continuity of Function

Problem Define g to be the function $g(z)=re^{\frac{i\theta}{2}}$ if $z=re^{i\theta}$ with $r>0$ and $-\pi<\theta\le\pi$, and $g(z)=0$ when $z=0$. Is $g$ continuous from ...
6
votes
3answers
266 views

A calculus question

On the interval $(0, \infty)$,the function $f \geq 0$,$f' \leq 0$, and $f'' \geq 0$.Prove that $\lim\limits_{x \to \infty} xf'(x) = 0$.
6
votes
1answer
76 views

How find this numbers $a,b$

Question let function $f(x)=ax^2+b$, find all positive real numbers $(a,b)$,such for any real numbers,then we have $$f(xy)+f(x+y)\ge f(x)f(y)$$ My try: since $$f(xy)+f(x+y)\ge ...
6
votes
1answer
80 views

Wrong use of function notation $f(n)$

I've recently read in a book about computational complexity theory: $$ O(f(n)) = \{g:\mathbb N \to\mathbb R \cup \{0\} : \exists \xi > 0,n_0\in \mathbb N\;\: g(n) \leq \xi \cdot f(n) \;\: \forall n ...