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

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1
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2answers
22 views

period of $\cos(x) + x - \lfloor x \rfloor$?

What is the period of $\cos(x) + x - \lfloor x \rfloor$? This is what I have done: $x = \lfloor x \rfloor + \{x\}$ $\cos(x)$ has period $2\pi$ $\{x\}$ has period $1$ so $\cos(x) + \{x\}$ should ...
2
votes
2answers
38 views

Graph of the function $y = 2 + (x + 1)^3$

I know that this function will have the behavior of $Y = X^3$ but as I will translate for this function $(Y = X^3)$? I do this: $$(x + 1)^3 = x^3 + 3x^2 + 3x + 3 \quad y = x^3 + 3x^2 + 3x + 5$$ But ...
1
vote
1answer
42 views

Find a function with the property, or prove it doesn't exist

Today, I encountered the following problem in my research. I'd like to find a function $f(x_1, x_2, \ldots, x_n)$ such that $$ 0 = \frac{d f(x_1, x_2, \ldots, x_n)}{d a}\bigg|_{c_1,c_2,\ldots,c_n} ...
1
vote
0answers
13 views

How would we generate a basis of sigmoidal functions?

I am trying to figure out how to generate a basis of sigmoidal functions. My issue is thus: there are several possible generating functions for a sigmoid (logistic curves, error functions, arctangent, ...
0
votes
1answer
13 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
8
votes
1answer
92 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 ...
-2
votes
2answers
42 views

Proving the equation has no root. [on hold]

How to show that for $a\in \mathbb R$, the equation $x^2+12a^2+4ax-8a+8=0$ has no root?
2
votes
0answers
72 views

Inverse of $f(x)=x+\sin(x)$

I'm attempting to find $$f^{-1}(x)$$ for the function $$f(x) = x + \sin{x}.$$ So far I've tried some simple algebraic methods as well as rewriting $\sin{x}$ as a power series. I'm not quite sure where ...
1
vote
1answer
26 views

Integral with an unknown function

I am trying to solve this integral $$ \int \frac{f(x)}{g(x)}\frac{\mathrm dg}{\mathrm dx}\mathrm dx $$ where $g$ is an unknown function of $x$, and $f(x)$ is a known function that can be integrated ...
0
votes
1answer
19 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 ...
-1
votes
2answers
34 views

For a function $f:S\to S$, if $f$ is injective, then $f\circ f\circ f$ is injective

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
467 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? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
0
votes
1answer
37 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, & ...
3
votes
0answers
24 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 ...
2
votes
1answer
36 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 ...
3
votes
1answer
30 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 $\partial D_1\cup\{\infty\}$, ...
0
votes
3answers
66 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 $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
1
vote
1answer
124 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?
-2
votes
2answers
57 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
3answers
146 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 ...
1
vote
3answers
66 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 ...
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 ...
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
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0answers
45 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 $\Bbb{A}$ ...
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 ...
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
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 ...
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. ...
-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
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
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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?
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 ...
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 ...
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
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|} ...
-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
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 ...
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
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 ...
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
53 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?
2
votes
2answers
45 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 ...
3
votes
2answers
68 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 ...