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

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4
votes
0answers
148 views

Lower semicontinuous and discontinuous everywhere real bounded function?

Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere? Such a function seems too likely to ...
2
votes
1answer
57 views

How does a special case prove a surjection?

I have a problem understanding the following proof that claims a surjection. The text is translated from a german university textbook by Luise Unger (pardon any translation errors by me, please). ...
0
votes
1answer
52 views

Dimension reduction in the Lotka-Volterra model

I'm not sure if I can post this here, but check this out. This is some text from Boccara's Modeling Complex Systems. The thing which confuses me is that there is a dimension reduction from 4 ...
1
vote
1answer
55 views

is $x = \frac 1y $ a function or a relationship?

A teacher says $x = \frac 1y $ is indeed a function, because it's the same as $y = \frac 1x$ But I don't think so, because no restrictions for $x$ , so $x$ could be zero but there is no solution for ...
0
votes
1answer
268 views

How can I tell/compare the asymptotic complexity of a function?

For something, like a quadratic I just take the highest degree and see if it is theta or big O or Omega of n, correct? So like 2n^2+2n+1 could be theta(n^2). What are the general ...
0
votes
1answer
45 views

Finding range of transformation of function from range of original

I'm asked to find the range of $y = f(x-2)+4$, if the range of $y=f(x)$ is {$y| -2 \geq y \geq 5, y \in R$}. How do I go about finding this? I have no idea where to even start. I'm doing the course ...
0
votes
0answers
32 views

Restricting binary relations by composing with an “inclusion binary relation”

If $X' \subseteq X$ then we may define an inclusion map $\iota : X' \to X$ where $\iota(x) = x$. One use of $\iota$ is that we can express the restriction of some $f : X \to Y$ to $X'$ as $f|_{X'} = f ...
1
vote
2answers
759 views

definition of limsup of a function

I already have some idea on what $limsup_{x\rightarrow\infty} f(x)=\infty$ means but I would like to hear other ideas from the mathstackexchange community. Maybe some intuition would be helpful.
2
votes
0answers
61 views

Topology Question Help

Let $X$ and $Y$ be sets and let $f\colon X \to Y$ be a function. (a) If there is a subset $V$ of $Y$ such that $f(f^{-1}(V))$ does not equal $V$, must there be a subset $A$ of $X$ such that ...
3
votes
2answers
86 views

Express $\sin\frac{\pi}{8}$ and $\cos\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$

I've been trying with no success expressing this functions. a) $\sin\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$ b) $\cos\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$ I've tried formula of the double angle ...
2
votes
1answer
39 views

Quesiton about functions

I'm embarrassed asking this question. I took a long break. main question is : cartesian coordinate system in $R^n$ space is shown as $(x_1,x_2 ...x_n)$. Show that for $1\le i\le n$ each ...
2
votes
4answers
99 views

Is one form of a function more 'true' than another?

Here's a function: $f(x) = \frac{x^2}{x}$ Now, if we were to look for the $0$ value, we would end up with a division by zero situation. By simplifying it to an equivalent function: $f(x) = x$, this ...
1
vote
0answers
136 views

Showing well defined and onto for a function

Problem: Define $\mathbb{Z}_{mn} \rightarrow \mathbb{Z}_m \times \mathbb{Z}_n$ by $f([x]_{mn})=([x]_m,[x]_n).$ Show that $f$ is a function and $f$ is onto iff $gcd(m,n)=1$ $(=>)$ Suppose $f$ is ...
2
votes
0answers
49 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
0
votes
1answer
33 views

function on a fixed length sequence of positive real number that induces lexicographic order

Let $S$ be the finite set of sequences of length $n$, whose entries are all real positive numbers. Can we define a function $f$ on $S$ such that the order $f$ induces on $S$ i.e $\le$ is the same as ...
1
vote
1answer
436 views

Antisymmetric Relation

Determine whether the relation R on the set of all people is antisymmetric. (a) a is taller than b. (b) a and b are born on the same day. (c) a has the same first name as b. ...
0
votes
1answer
36 views

Function domain and range

I have a function defined as: $ f: \mathbb{N}^3 \rightarrow \mathbb{N}$ with signature $f(x_1, x_2, x_3)$, no more. I need to do a certain proof. What does the definition mean? For some reason, I ...
0
votes
2answers
98 views

Is $L_\infty$ norm the smallest or largest?

I am a little bit confused. For a $L_p$ function norm, is it true that for any $ p<\infty $, $$ \|f\|_p>\|f\|_\infty$$ Is the statement true for any domain? I want to know more inequality about ...
2
votes
0answers
51 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
0
votes
2answers
59 views

Minimize $ab+bc+ca$ under three second degree constraints

As stated in the title, my problem is quite simple. Minimize $ab+bc+ca$ under these three constraints: $$ a^2+b^2=1 $$$$ b^2+c^2=2 $$$$ c^2+a^2=2 $$ I can brute force it, with some intelligence of ...
0
votes
2answers
47 views

finding the inverse of function. the domain is given and the question requires to find the inverse.

Find the inverse of the function. $$f(x)=x^2 + 4x$$ Domain: $x \geq -2$ I have done: $X = y^2 + 4y$ $X - 4y = y^2$ $\sqrt{X - 4y} = y$ My answer: $f^{-1}(x) = \sqrt{X - 4x} $ Real answer: ...
0
votes
1answer
85 views

The ring structure of the set of all functions from a finite field $K$ to itself

$K$ is a finite field and $F$ is the set of all functions of $K$ in $K$. In $F$ we define $(f+g)(a)=f(a)+g(a)$ and $(fg)(a)=f(a)g(a)$. Show that $F$ is isomorphic a $k[x]/I$ for a some ideal ...
1
vote
1answer
87 views

Unbounded function with uniform bounds on Integrals

I'm looking for examples of unbounded functions $f\colon\Bbb R\rightarrow\Bbb R$ with the property that for all $L$ exists some $C(L)$ with $$\int_x^{x+L} f(t)dt \le C$$ for all $x$.
1
vote
5answers
74 views

Function notation in words

My function is $f(x)=x(x-4)(x-2)(x+2)$ and I need to know how to put it into words. My math teacher told me how to put it in words and I've completely forgotten. I've looked everywhere and can not ...
4
votes
1answer
101 views

For bijection $f:A \rightarrow B$, prove that $f^{-1} \circ f = {\text {id}}_{A}$

I have to prove that for a bijection $f:A \rightarrow B$, $f^{-1} \circ f = {\text {id}}_{A}$, where ${\text {id}_A}$ is the identity function of $A$, and we define $f^{-1}: B \rightarrow A$ by ...
12
votes
3answers
1k views

Why is $\tan(x)$ a function?

A function $f:X\rightarrow Y$ maps each $x\in X$ to some $y \in Y$. So consider $\tan{\frac{\pi}{2}}$ for which $\tan(x)$ is undefined, so in this case, $\tan(x)$ does not map to an element of its ...
0
votes
2answers
28 views

The Image of the Function: $f$:

I am a bit confused how they arrived at this answer: The image of the function $f: \mathbb N \to \mathbb R , f(n) := \frac{(-1)^n+1}{3} $ The answer they got was $\{0, \frac{2}{3}\}$ Can someone ...
2
votes
1answer
261 views

Lebesgue-integrable function

I have to decice if the following function is Lebesgue-integrable on $[0,1]$: $$g(x)=\frac{1}x\cos\left(\frac{1}x\right) $$ where $x\in[0,1]$.
0
votes
3answers
143 views

$f(x+y)=f(x)f(y)$ [closed]

Let $f$ be a function from the positive integers to the positive integers that satisfies the property: $$ f(x+y)=f(x)f(y) $$ for all pairs of positive integers $(x,y)$. If we are given that ...
1
vote
2answers
76 views

Say I have a function $f(x) = x^2$. Can this be a surjective function?

If I'm getting the unto and 1-1 concepts right, $f(x) = x^2$ is always 1-1 as $x$ always maps distinct objects in codomain ($x^2$). But it's not a surjective function since you can't get all $x$ in ...
7
votes
3answers
259 views

Show that $\frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )}$ is decreasing in $y > 1 $.

I am interested in the function $f(y) = \frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )},$ for values of $c \in (0,1)$, and $y > 1$, and have been trying to show that the function is decreasing. I ...
0
votes
1answer
105 views

Is there a function that is surjective if and only if it is injective?

Apparently there are some but I can't think of any.
1
vote
2answers
622 views

Suppose $g$ is even and let $h=f \circ g$. Is $h$ always an even function?

I came across one of the following problems in my homework set: $$ \text{Suppose} \, g \, \text{is even and let} \, h=f \circ g. \text{Is} \, h \, \text{always an even function?}$$ I came to the ...
2
votes
2answers
34 views

if $f(k/N)\rightarrow0$ as $N\rightarrow\infty$ for any $k$, must $f(h)\rightarrow0$ as $h\rightarrow0$?

If $f$ is a function defined on [$\mathbb{R}$ or $\mathbb{C}$] such that for any [real or complex] $k$, $f(\frac{k}{N})\rightarrow0$ as $N\rightarrow\infty$ in $\mathbb{N}$, must it be true that ...
4
votes
1answer
124 views

Function Surjectivity Proof

I have this question: Prove that a function $f:X\rightarrow Y$ is surjective iff for any finite set $Z$ and any function $g:Z\rightarrow Y$ there exists a function $h:Z\rightarrow X$ such that ...
1
vote
1answer
310 views

Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
0
votes
2answers
107 views

Prove that $f$ is an increasing function

Let $f$ be defined on the open interval $(a,b)$ and assume that for each $x\in(a,b)$ there exists a $1$-ball $B(x)$ in which $f$ is increasing. Can someone help me to prove that $f$ is increasing ...
0
votes
1answer
37 views

Find an Inverse function

I need to find the inverse of those functions: $x \mapsto \sin e^{x}$ $x \mapsto e^{\sin x}$ I know that the way is to solve the equation $y = f(x)$ for $x$, and I did it with functions like ($x ...
1
vote
2answers
555 views

Opposite function definition

How would I define the opposite of function. If for example I have the function F(x) = y how can use this function to define the function ...
-1
votes
1answer
78 views

function inside function - maximum value

I am trying to calculate my algorithm efficiency as I posted in here. One of the things that I need to do is to find the maximum value between 1 to $G$ that belongs to $$f(x) = 2f(x - 1) - 1; F(0) = ...
1
vote
1answer
44 views

Does there exist an injection from $P(S)$ to $u(S)$

Let $S$ be an uncountable set , let $u(S)$ denote the set of all uncountable subsets of $S$ and let $P(S)$ denote , as usual , the powerset i.e. the set of all subsets of $S$, then does there exist an ...
10
votes
2answers
288 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.
0
votes
1answer
35 views

Please help me isolate this variable

I need to isolate 'B' in the following equation: $$ E\left( \left({A-B\over C}\right)^2 -D^2 \right) = B(1-B) $$ Can anyone help? Is it possible?
0
votes
1answer
66 views

Number of functions in set

I'm studying for my discrete math midterm and having some trouble with this question: A{1,2,3,...,n} and B={a,b,c} A fixed integer k such that 0 =< k =< n and a fixed subset S of A having size ...
0
votes
1answer
330 views

Composition of Periodic Functions.

Suppose $f(x)$ is defined for all $x$; then $f \circ \cos x$ is periodic and $2 \pi$ is a period. Conversely, if $g(x)$ is defined for all $x$ and is periodic, with $2 \pi$ as a period, can one find a ...
0
votes
3answers
62 views

Show function $f(x)=\frac{r^2 \cdot a}{a^2+x^2}$ is strictly decreasing

I want to show, that $$f(x)=\frac{r^2 \cdot a}{a^2+x^2}$$ with $x,a,r \in \mathbb{R}$ is strictly decreasing for $x>0$ A function is called strictly decreasing, if $x_1<x_2 \Rightarrow ...
0
votes
1answer
131 views

Discontinuity in multivariable calculus

Show that the function F(x, y) = x(1 - cos(x - y))/(x y)^2 has removable discontinuity along the line x = y. Which values should we assign to this function on the diagonal x = y in order to turn it ...
0
votes
2answers
77 views

Is the image of the difference equal to the difference of the images?

Let $f: A → B$ be a function; $E, F ⊆ A.$ Is $f(E - F) = f(E) - f(F)$ ?
2
votes
1answer
288 views

Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ ...
4
votes
1answer
406 views

Set-builder notation function definition

I know that a function is a subset $f \subseteq X \times Y$ such that \begin{eqnarray} \forall x \in X, \exists ! y \in Y | (x,y) \in f \end{eqnarray} First, is it possible to express what a ...