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

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8
votes
1answer
212 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}. ...
0
votes
1answer
16 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
0
votes
1answer
21 views

Proving question on Functions.

For a function $f:S\rightarrow S$ , if $f$ is injective, then ,$f\circ f\circ f$ is injective. Can I get hints on how I can prove that it is true or false?
7
votes
4answers
439 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
2
votes
3answers
17k views

If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that…) [duplicate]

Possible Duplicate: Injective and Surjective Functions If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one given that $f$ is a function from A to B and $g$ a function from B ...
2
votes
1answer
34 views

Show that if $f$ is a proper,surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
0
votes
0answers
29 views

If $T^{k}=0$ of some $k$, then $T^n=0$, where $n$ is dimension [duplicate]

Let $V$ an $n$-dimensional vector space and $T$ a linear operator on $V$. Suppose that there is some positive integer $k$ such that $T^{k}=0$. Prove that. $T^{n}=0$
0
votes
1answer
35 views

Good Triple Well Function [on hold]

I am looking for a good triple-well function with good control over the barrier height. Let's say that $y=9x^{2}-6x^{4}+x^{6}$. In this function there are three wells (even though only two wells are ...
1
vote
1answer
35 views

Proof that a function is unbounded [on hold]

I have this function \begin{equation*} f(x):=\left\{\begin{array}{cl} \frac{3}{2}x^{\frac{1}{2}}(\sin\frac{1}{x})+x^{\frac{3}{2}}(\cos\frac{1}{x})(-x^{-2}), & \mbox{for }0<x\leq 1,\\ 0, & ...
1
vote
3answers
64 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
3
votes
0answers
19 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k^n}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
3
votes
1answer
27 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
1
vote
2answers
18k views

Proving Functions are Surjective

Prove the following are surjective, or disprove with a counter-example: $f\colon \mathbb{Q} \to \mathbb{Q}$, $f(x) = 1 + 2x$. $f\colon \mathbb{Z} \to \mathbb{N}\cup\{0\}$, $f(x) = |1 - x|$. ...
2
votes
1answer
284 views

If $f \circ g$ is onto then $f$ is onto and if $f \circ g$ is one-to-one then $g$ is one-to-one

I am trying to make a picture in my head so I can understand and remember the rules. So if $f \circ g$ is onto, it is onto because the function $f$ maps every element from a set $B$ to a set $C$ ...
0
votes
1answer
33 views

Proving composition of functions [duplicate]

I am trying to prove the following theorems: Let A, B, and C be nonempty sets and let $f : A \rightarrow B$ and $g : B \rightarrow C$. If $g \circ f : A \rightarrow C$ is an injection, ...
0
votes
3answers
63 views

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective.

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective. My attempt: Using contrapositive, if Y is not injective. then Y ∘ Y is not injective, the there exist x, x' ∈ X with x ≠ x' but ...
-4
votes
1answer
25 views

Find the domain and range of $f$ and $f^{-1}$ [on hold]

Find the domain and range of $f$ and $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
-2
votes
2answers
54 views

Proving question on sets

I am unable to understand this question. I have to say whether its true of false and then prove it but I can't proceed with the question unless I understand it. Let $S$ be the set of stars in our ...
1
vote
1answer
123 views

Shoud a function be defined for all elements in the domain in order to be surjective/bijective?

In other words, the surjection says: for any y in the codomain there should exist x in the domain. Now, do I need for every x in the domain to have an y in the codomain for surjectivity?
1
vote
1answer
48 views

Does proper map $f$ take discrete sets to discrete sets?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true? $1$. The map $f$ takes discrete sets to discrete sets. $2$. If $f$ is ...
1
vote
3answers
145 views

Limit of a Sine Function

Calculate via the limit definition: $$\lim\limits_{x \to \frac {\pi}2^-} \frac{\sin^2(\frac {\pi}2-x)}{\sqrt{\pi-2x}},$$ I tried to calculate this limit using the definition of a limit and got ...
6
votes
1answer
48 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
0
votes
1answer
18 views

How should one define the cross product for two vector valued functions?

In many textbooks regarding vector analysis, the cross product is mention in the section about the properties of the derivative of vector functions, but there seems to be no explanation what that ...
12
votes
0answers
141 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
1
vote
1answer
21 views

Characteristic Function in the subset E

Let $E \subseteq \mathbb R$. Then the characteristic function $\chi_{E}:\mathbb R \to \mathbb R$ is continuous if and only if a) $E$ is closed. b) $E$ is Open. c) $E$ is both Open and Closed. d) ...
0
votes
1answer
23 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...
1
vote
0answers
44 views

Let $f(x)$ be a function that satisfies $f(f'(x))=3xf(x) \iff x\in \Bbb{A}$. Find $f(x)$. [on hold]

I had this question on an exam and I didn't even know how to start. Could anyone give me some hints? Let $f(x)$ be a function that satisfies $f(f'(x))=3xf(x) \iff x\in \Bbb{A}$. Where $\Bbb{A}$ ...
0
votes
0answers
14 views

Help in identifying the one dimensional map

In the paper: http://inds08.uni-klu.ac.at/INDS2008/INDS08_System_Identification_using_Symbolic_Chaotic_Sequence.pdf there is a chaotic map in Eq(11) $$c_{n+1} = \frac{\gamma c_n(1-c_n^2)}{1+\rho ...
1
vote
3answers
51 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
1
vote
1answer
27 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
-6
votes
0answers
23 views

Linear relationships [on hold]

The function rule $C = 10n + 26$ relates the number of people $n$ who attend a small concert to the cost $C$ in dollars of the concert. Make a table of input and output pairs: show the cost if $27$, ...
1
vote
1answer
36 views

Prove that $f$ in monotonic

In my assignment I have to prove the following: Let $f$ a continuous function in $\Bbb R$. Prove the following: if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R. ...
1
vote
2answers
24 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
1
vote
1answer
26 views

What does “2- place real function” mean?

What does "2-place real function" mean? This comes up in the context of copulas, as here.
1
vote
2answers
22 views

Derivatives problem

Given the equation $f(x)=\frac{2x+4}{\sqrt{x}}$, evaluate $f(0.5)$ and $f'(0.5)$. I am having a problem understanding the problem. The first part is straight forward, but it's the second part I'm ...
0
votes
2answers
41 views

If $f(z)= \frac 1z $ be defined and analytic on region $ |z| \gt 1 $ in $ \Bbb C $ then can we find an entire function $g$ such that :

$g$ should be such that $f(z)=g(z)$ on $ |z| \gt 1$ in $\Bbb C $. Now,Can we plainly apply uniqueness theorem and say that such a function $g$ can not exist?
-3
votes
2answers
48 views

Prove $f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) \cap \dotsb \cap f(A_n)$ [on hold]

Let $f: R \to R$ be a one to one function. For any collection of subsets $A_1, A_2, A_3, \dotsc A_n$ of $R$, prove that $$f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) ...
2
votes
2answers
39 views

Proving a real valued function is periodic, and sketching it using obtained information

Consider an arbitrary function, something like $f\left ( x \right )=\arccos \left ( \sin \left ( 4x \right ) \right )$. Its graph looks like this: I was greatly confused by the image below, because ...
0
votes
2answers
45 views

Is the function continuous - $f(x) = \frac{1}{\sin x} + \frac{1}{x-1}$

I have an assignment in which I have to prove that a function "recieves every real value, where $x\in (0,1)$". Here is the function: $$f(x) = \frac{1}{\sin x} + \frac{1}{x-1}$$ I don't know the ...
0
votes
2answers
36 views

Is this definition correct for the inverse of a function?

Is this definition correct for the inverse of a function? Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the ...
2
votes
1answer
36 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
1
vote
1answer
20 views

Inverse of a set of ordered pairs.

An exam ask me the following question. Let $r=\{(x,y) \ | \ x \in [-1,1] \ \text{and} \ y=x^2\}$, is the following statement true? $$r^{-1}=\{(x,y) \ | \ x \in [0,1] \ \text{and} \ y=\pm\sqrt{|x|} ...
2
votes
5answers
40 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in ...
2
votes
1answer
309 views

Function that is not differentiable at a point

I am looking for a continuous function to be used in fourier series graph that have the same value at both $-\pi$ and $\pi$ but has a very poor differentiability at a point. I have one: ...
0
votes
2answers
35 views

The functional equation $x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=…$

Consider the functional equation $$x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=...$$ The equality continues to infinity. Is there $C(x)$ that satisfies all the equality? If there is, what is it? ...
2
votes
4answers
53 views

If $f$ and $g$ are both functions from $X$ to $X$ and $f\circ g$ is the identity function, does $g\circ f$ also have to be the identity map?

If $f$ and $g$ are both functions from the set $X$ to $X$ and $f\circ g$ is the identity function, does $g\circ f$ also have to be the identity map? How (if at all) does your answer change if $X$ is ...
0
votes
2answers
27 views

Definition of a function of sets

Suppose I have two sets of variables: $S_1 = \{X_1,X_2,\dots\}$ and $S_2 = \{Y_1,Y_2,\dots\}$. I want to define a function $f$ that takes all variables in $S_1$ and $S_2$ as parameters: ...
0
votes
3answers
52 views

Why is $f:\mathbb{R^+}\to \mathbb{R}$ defined by $f(x)=x^2$ not an invertible function?

Why is $f:\mathbb{R^+}\to \mathbb{R}$ defined by $f(x)=x^2$ not an invertible function? I know the answer: because it's not onto, but what is problem with it? what does it break the invertibility? ...
1
vote
1answer
32 views

Exceptions in functions

I have recently started studying functions(topics such as periodicity, odd/even, into/onto, etc.). I was wondering if there are any strange exceptions to the general rule that is taught?
3
votes
2answers
67 views

When is the function Continuous?

In my assignment I have to determine when is the function continuous. This is the function: \begin{equation} g(x) = \begin{cases} \left\lfloor {\sin\frac{1}{x}}\right\rfloor&\text{if} \space ...