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

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0
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1answer
15 views

Algorithmic question regarding permutations

An algorithm i'm reading has the first step saying. The algorithm is a path enumeration algorithm which puts all permutations into lexicographic order. The algorithm works as follows Input: A ...
1
vote
2answers
16 views

Find the image of $A=(-2,1) \times [-2,2)$ under the function $f(x,y)=x^2y$

I have function $f(x,y)=x^2y$ and I have to find image $f[A]$ where $A=(-2,1) \times [-2,2)$ we have that $-2 < x < 1$ and $-2\le y<2$ $0 \le x^2 < 4 $ I claim that the image of ...
1
vote
1answer
25 views

What is this function?

In python I made this function: def f(x): eqStr = '' for y in range(int(x)): eqStr += 'x**%s + ' % (y) eqStr += '0' return eval(eqStr) ...
0
votes
0answers
9 views

Is there any function analogous to Heaviside step function for boolean input?

I want to have a mathematical model in an analogous way with step function (for integers). To be more specific, I want a function (in theory, not in any programming language) such as function ("i same ...
-1
votes
1answer
24 views

Let f be a map from R to R.Show that f’(a) is the derivative of f at a…read the question [on hold]

Let $f$ be a function from $\Bbb R$ to $\Bbb R$. Show that $f’(a)$ is the derivative of $f$ at $a$ if and only if $$\lim_{h\to0}⁡\dfrac{f(a+h)-f(a)- f' (a)h}{|h|}=0.$$
2
votes
1answer
24 views

Fixed point of a mapping

How to prove that every continuous $f:S^1 \to S^1$ such that $deg(f)\neq 1$ has a fixed point? One hint is that if $f(x)\neq x$ for any $x\in S^1$ then $f$ is homotopic to the antipodal map $a$ but I ...
0
votes
0answers
33 views

Why does Power Set represented by $2^x$ takes only $0$ or $1$ as values for $x$

While I was studying about Function Spaces I've seen an example of Function Space from function space of Power Set that tells that it(power set) maps from $X$ to $\{0,1\}$. I couldn't get how that ...
1
vote
0answers
7 views

Space $C^2(\overline{U})$ for open set $U$

Let $U$ be a bounded open domain in $\mathbb{R}^n$. Does the space $C^2(\overline{U})$ (the bar over $U$ means closure) mean the set of twice-differentiable functions $u$ such that $u, u_t, u_{x_i}$ ...
0
votes
2answers
18 views

Define f : Z/3Z → Z/3Z by f ([a]) = [2a + 1].

For this problem, I have to prove the function is well-defined, is surjective, and is injective. For seeing it is well defined, I have this: Assume [a1] = [a2] in the set of equivalence classes Z/3Z. ...
-1
votes
0answers
18 views

Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b).

So, as stated above, here is my question: Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b). First, I have to prove that ~f is an equivalence relation on A. So I need to show that ...
0
votes
0answers
16 views

Bijective functions on a finite set

Suppose that A is a finite set and f : A → A and g : A → A are functions. I need to prove that g ◦ f is a bijection if and only if f and g are bijections. So, could I say: Assume g of f is a ...
0
votes
1answer
23 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
1
vote
2answers
34 views

Finding a counterexample to a function proof

This is my proof: If f and g are surjective, then g ◦ f is surjective, with f: A $\to$ B and g: B $\to$ C. I have successfully proved this, but now I have to disprove the converse by finding a ...
1
vote
1answer
18 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
1
vote
1answer
19 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
votes
1answer
47 views

Why do we have trigonometric functions besides $\sin(x)$?

Probably a terrible question, but I've been curious and can't come up with a reason besides convenience for myself with my limited knowledge. Why do we have $\cos(x)$, $\tan(x)$, etc. when all of ...
4
votes
2answers
50 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
-1
votes
0answers
38 views

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ is no-where differentiable on $\mathbb R$?

Does there exist a real valued differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ , the derivative of $f$ , is no-where differentiable on $\mathbb R$ ?
0
votes
0answers
19 views

Integrable function and measure set relation

There is this problem that I am stuck with. I would be very thankful if someone gave me a hint about how to do it. Please do not give full solution. $A$ is a rectangle in $\Bbb R^k$ and $B$ is a ...
-1
votes
0answers
27 views

Define $f : Z/4Z → Z/4Z$ by $f ([a]) = [3a + 1]$.

Define $f : Z/4Z → Z/4Z$ by $f([a]) = [3a + 1]$. (a) Prove that $f$ is a well-defined function. (b) Prove that $f$ is surjective. (c) Prove that $f$ is injective. I'm having trouble with this ...
-3
votes
2answers
44 views

True or False. If f(-2)=2 then x+2 is a factor of f(x) [on hold]

True or False. If f(-2)=2 then x+2 is a factor of f(x). Explain your answer. Explain Using the remainder Theorem I believe the teacher said. For college algebra.
-2
votes
1answer
32 views

What means $A \subsetneq X$ with A ~ X? [on hold]

How it is possible to have a subset A, which is $\neq$ to X and at the same time they have an equivalence relation ~? When $A \subset X$ therefore a $\in$ A is also a $\in$ X. With A ~ X ...
1
vote
2answers
32 views

Proof of uniform continuity of a rational function

Let $f\colon [0, \infty) \to \mathbb{R}$ be defined by $f(x)= \dfrac{4x}{1+x}$. Show that $f$ is uniformly continuous on $[0, \infty)$.
3
votes
2answers
78 views

Circle to circle homotopic to the constant map?

How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$ (a) has a constant point and that (b) $f$ maps $x$ to its antipodal point $-x$?
0
votes
2answers
35 views

finding the inverse function of $f(x)=x+\frac{1}{x}$

find the inverse function of $f:\Bbb{R} \to \Bbb{R}$ where $f(x) =x+\frac{1}{x}$. I have tried raising to the power of $2$ but it did not work.
3
votes
2answers
267 views

Mapping the open ball to itself?

How to prove that there exists a continuous function $f:B^2 \to B^2$ without constant points? Here, $B^2$ is the unit open ball. I guess $f$ can be for example like this $f: re^{iax} \to re^{ibx} $ ...
0
votes
0answers
7 views

Project a function on a space?

The problem I'm solving is $\begin{cases} & \dot{x}_{1} = -u \\ & \dot{x}_{2} = 4x_{1} \end{cases}$ $x_{1}(0) = x_{2}(0)=0, |u| \leq 3, t \in [0;2], u^{0}(t)\equiv0, J[u] = -2x_{1}(2) + ...
0
votes
0answers
17 views

How can a function with asymptotes be defined as a mapping?

A mapping takes each element of a set S and associates it with an element t in some other set T. I believe functions to be mappings. Yet we happily call such as $\frac {x^2}{x+1}$ a function, even ...
2
votes
2answers
41 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
5
votes
2answers
168 views

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$.

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$. My answer : Suppose $f$ is injective and $f(x) \in ...
0
votes
1answer
24 views

How to find $f^{−1}([9,0])$ and $f([1,4])$ for $f(x)=x-6\sqrt{x}$?

$f$ is a the function defined by $$\eqalign{ f\colon& \Bbb R &\rightarrow \Bbb R_+\\ & x&\mapsto x-6\sqrt{x} }$$ Find $f^{−1}([-9,0])$ and $f([1,4])$.
3
votes
4answers
55 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
3
votes
1answer
22 views

Specific piecewise-function SAT2 question

Taken from Barron's SAT Math Level 2 prep book: If f(x) = i, where i is an integer such that i ≤ x < i + 1, the range of f(x) is ...
1
vote
2answers
31 views

Condition on $ a$ and $b$ so that $f(x)$ has a root?

Let $f(x) = ax(1-bx^{2/3})-1$ where $a$ and $b$ are positive. What is the necessary and sufficient condition on $a$ and $b$ such that $f(x)$ has at least one real root?
0
votes
0answers
16 views

A doubt in Kreyszig's Functional analysis

As an application of the Uniform boundedness Theorem, it is proved in Kreyszig's Functional analysis that "There exist real valued continuous functions whose Fourier series diverge at a given point". ...
0
votes
1answer
24 views

Exercises about function composition, bijection and inverses.

So I'm gonna have a test about: Definition of a function, of a surjective and injective function, inverse function, proof that if a function has na inverse, then the functions is bijective, etc The ...
2
votes
1answer
34 views

The supermum of E

Let $f\ [0,1]\longrightarrow [0,1]$ be increasing function. let: $$E=\{x\in [0,1] \mid f(x)\geq x \} $$ Show that $E$ has a supermum $b$ and that $f(b)= b$. we have $x\leq 1$ since $f$ is ...
-1
votes
1answer
21 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
0
votes
0answers
10 views

Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
1
vote
2answers
41 views

If $f'(x)$ has a limit as $x\to x_0$, then the function $f$ is differentiable at $x_0$

I've got a question about mathematical analysis of one-variable functions. Assume that we have a function defined for $x \neq x_0$ as composition/sum/product of differentiable functions and also ...
0
votes
1answer
24 views

How many function A to B satisfied from f(1)=x

What does it mean to satisfy a function A to B from $f(1)=x$ ? Where $$ A=\{1,2,3,4\}\ \ \text{and}\ \ B=\{x,y,z\}$$ The answer should be $3^3$, but why?
1
vote
2answers
48 views

Confusion in notation of functions.

Let us consider the following notations for $x \in X,y\in Y ,z \in Z$. $$F(x,y,z)=x^yy^z$$ $$F_x(y,z)=x^yy^z$$ I am clear with former notation , but I saw latter one too , what's the difference ...
0
votes
2answers
36 views

Let $A$ and $B$ be countable sets. Is there any function $f$ such that a certain condition holds for an uncountable number of functions $g$?

Let $A$ and $B$ be countable sets. Is there any function $f:A\to B$ such that there exists uncountably many functions $g:B\to A$ such that $g\circ f=\operatorname{id}$ but $f\circ ...
0
votes
3answers
51 views

Come up with this function

Here's a fun math question: Come up with a function where $$ \begin{align} g''(1) &= 0 \\ g(0) &= 0 \\ g'(0) &= 0 \\ g(1) &= 1 \\ g'(1) &= 1 \end{align} $$ I've tried multiple ...
1
vote
1answer
19 views

Extension Theorem of twice continously differentiable functions?

Is there a theorem which guarantees me that any function $f$ with bounded first and second order derivatives defined over a compact interval of $\mathbb{R}^2$ can be extended to a twice continously ...
2
votes
1answer
56 views

How can I write a function like this? [on hold]

I need to write down and use a function which looks like this. It is some kind of a sinus function. I've no idea how this function looks like, and that is the reason I am looking for your help. Thanks ...
0
votes
2answers
15 views

Function bijective proving.

Let $\mathbb{C}$ be the set of all complex number. $z\in \mathbb{C}$ Given a function $$ f : \mathbb{C} \to \mathbb{C} $$ $$f(z) = (1+2i)z+5i$$ Prove that it is bijective. First, prove ...
0
votes
2answers
54 views

Closeness of set for not everywhere continuous function

I have a function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ where $w(x)\in[0,2\pi)$. I am also given for free that $w$ is continuous on $\mathbb{R}^2\backslash\{(x_1,0)\mid x_1\ge0\}$. I ...
1
vote
5answers
46 views

Linearity of a function.

I am requested to determine wether these functions are linear or not; to do that, I've to verify both the necessary conditions that are: $f(x+y) = f(x) + f(y)$ $f(\alpha x) = \alpha f(x)$ Now, my ...
4
votes
2answers
63 views

Existence of a differentiable function $f$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$ ?