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

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3
votes
3answers
62 views

$f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at x=1

How to prove that $f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at $x=1$ ? Ya,even I used intuition at ...
1
vote
4answers
64 views

If $f(f(x)) = x $ has at most 1 solution, then so does $f(x) = x$.

Let f be defined on [0,1] and its values are between 0 and 1. If $f(f(x)) = x $ has at most 1 solution, then $f(x) = x$ has at most 1 solution. Please, give me a hint how to prove this.
-2
votes
2answers
46 views

Example of periodic $f\left(x\right)+xg\left(x\right)$, where f is even function and g is periodic

Let f and g be non-constant functions defined on $\mathbb{R}$. Let f be even function and g be periodic one. I need an example of functions f,g such that $f\left(x\right)+xg\left(x\right)$ is ...
1
vote
0answers
32 views

Easily computable function that is one-to-one and onto with 2 or more inputs

I am looking for a function that will take take n inputs and have 1 output, and knowing the 1 output and function you can get back the n inputs, without error and computationally easy in both ...
0
votes
0answers
21 views

pre-image of intersection

This is a basic question and I know this is typically not how this is proven but I was wondering if the following is a valid proof of showing that given two disjoint sets, say V and W in the ...
0
votes
0answers
10 views

Tools to Find Limits of Function Outputs from Functions and Input Limits

I've got a complicated (to me) series of functions and I need to determine the limits of the outputs based on various assumptions about the inputs. I took a quick look at SymPy, but it does not seem ...
8
votes
4answers
680 views

Is the following a valid mathematical statement?

For all $f:\mathbb N\to\{1,2,3,\ldots,100\}$, If $f$ is a one to one correspondence, Then $f^{-1}(2)=3$ It seems as though this should not be a valid statement, since the implication fails to ...
0
votes
1answer
25 views

Inverse of function containing modulation and flooring

I have a function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined as: $$f(x) = ((x \bmod 9) + 1) \cdot 10^{\lfloor \frac{x}{9} \rfloor}$$ It seems to be injective, but I'm not sure about it being ...
1
vote
2answers
26 views

Show that a zero of $f$ is a fixed point of $g$

I want to show that a solution of the equation $x^2+cos(x)-10x=0$ is a fixed point of $g(x)=(x^2+cos(x))/10$. I tried using the quadratic equation but my solution doesn't simplify nicely in $g$. I'm ...
2
votes
4answers
41 views

Derivative of Function with Rational Exponents $f(x)= \sqrt[3]{2x^3-5x^2+x}$

I have a question following: $$f(x)=\sqrt[3]{2x^3-5x^2+x}$$ Here's what I did, $$f(x)=\sqrt[3]{2x^3-5x^2+x} \\ = (2x^3-5x^2+x)^{3\over2} \\\\f'(x) = {3\over 2}(2x^3-5x^2+x)^{3\over2}(6x^2-10x+1)$$ ...
0
votes
0answers
12 views

common symbolic notations for multi-valued and bag-valued functions

What are the commonly used notations for a multi-valued $f_m: A\rightarrow B$, and bag-valued $f_b:A \rightarrow B$ functions from A to B? To be more precise: $f: A \rightarrow B$ is usually used to ...
1
vote
2answers
23 views

Show uniqueness of solution to a function with infinite limit (and sequences).

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics, so Lipschitz continuity isn't part of the course (to my knowledge). ...
0
votes
2answers
14 views

Applicability of the definition of limit for a constant function

I am giving the following definition of limit (Chiang, Fundamental methods of mathematical economics): As $v$ approaches a number $N$, the limit of $q=g(v)$ is the number $L$, if, for every ...
1
vote
3answers
56 views

Finding the Range of a Trigonometric function

The range of $$f(x)=3\cos^2x-8\sqrt3 \cos x\cdot\sin x+5\sin^2x-7$$is given by:(1)$[-7,7]$(2)$[-10,4]$(3)$[-4,4]$(4)$[-10,7]$ ANS: (2) My Solution The equation can be written as: $$3\cos^2x-8\sqrt3 ...
0
votes
1answer
25 views

Does “f : A → B” need to be one-to-one and onto so that if Y ⊆ B, then the inverse image of Y under f and the image of Y under f-1 are equal?

I was solving a problem in section 5.4 of "How to Prove it Right" by velleman. Below are the problem and my answer. According to my inspection, $f$ didn't need to be one-to-one and onto. Did I miss ...
0
votes
1answer
22 views

Does $f(X \setminus A)\subseteq Y\setminus f(A), \forall A\subseteq X$ imply $f$ is injective ?

I know that if $f:X\to Y$ is injective then $f(X \setminus A)\subseteq Y\setminus f(A), \forall A\subseteq X$ . Is the converse true i.e. if $f:X \to Y$ is a function such that $f(X \setminus ...
0
votes
2answers
53 views

How to compute probability related to a difference of two random variables

I am studying Joint Probability Distributions and Random Samples. I have a function for a probability distribution, defined as: $ f(x, y) = K(x^2 + y^2)~~~~~~~~~ 20 \leq x \leq 30, ~~~20 \leq y ...
-1
votes
0answers
16 views

need help adjusting the calculation result for a chart

First of all, I'm really sorry as to most of you this question will look elementary. But to me it's not that simple. I'm calculating some sort of a horizontal bar chart for 6 products of different ...
2
votes
1answer
69 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
2
votes
3answers
46 views

$f(x, y) = x - y$ - Injective?

Given the following function on $\mathbb{Z}$: $x,y \in \mathbb{Z}: f(x, y) = x - y$ As I understand, this function is surjective, i.e. each element of $\mathbb{Z}$ is the image of at least one ...
0
votes
2answers
20 views

Substitutions as mappings from the set of Propositional Variables to the set of Formulas

Rautenberg defines substitutions in propositional calculus as follows: " A (propositional) substitution is a mapping σ : PV →F that is extended in a natural way to a mapping σ : F → F " PV: set of ...
1
vote
1answer
46 views

Prove that $f_n(x)$ converges uniformly on $[1,\infty)$ but not on $(0,1)$.

Let $f_n(x) = \frac{\sin (nx)}{nx}$. Prove that $f_n(x)$ converges uniformly on $[1,\infty)$ but not on $(0,1)$. Let $x\in [1,\infty)$. We observe that: $$\frac{1}{nx} \le \frac{1}{n} \to 0$$ So by ...
1
vote
1answer
33 views

What implies $f_n (y) \leq f(x) + \epsilon$ about $f$ ?

Let $X$ be a regular topological space. Question: For which functions $f : X \rightarrow \mathbb R$, can we find a sequence of functions $f_n : X \rightarrow \mathbb R$ such that: $\forall ...
0
votes
2answers
24 views

Derivative of implicit fuction

I want the proof of implicit fuction derivative. I don't know why I should calculate derivative of all monomials towards x for finding $y'$ (derivative of $y$) with respect to x at equations such as ...
0
votes
1answer
24 views

Limit of two functions

Can we prove that if two functions equal to each other in any finite time, then the infinite limit would be the same?
3
votes
3answers
39 views

Graph and domain of $\frac{2}{7+\sqrt{x}}$

How to sketch the graph of $\frac{2}{7+\sqrt{x}}$? Can anyone give me some hints ?
0
votes
0answers
17 views

Simple function design: asymptotic to $k$ and concavity condition

Referring to the answer given by user lhf to this question, I was wondering whether that function, $f(x)=ae^{bx}$, may suit my needs if I had to add additional features. These features are: ...
0
votes
1answer
31 views

If $f$ and $g$ are monotone increasing and if the composite function $f\circ g$ is defined, then it is also monotone increasing.

For this question the definition for monotone increasing is that $f$ is monotone increasing if for all $x_1<x_2$ where $x_1,x_2\in I$, $f(x_1) \leqslant f(x_2)$. I have to apply that definition to ...
1
vote
3answers
42 views

Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$
2
votes
1answer
21 views

Interval functions

I'm kinda a greenhorn in maths and young and unexperienced, but one thought popped in my head that google couldn't satisfy. Let $g(x) = \sin(x)$. Then wouldn't be $g([a,b]) = \{ (x, y): a \leq x ...
8
votes
8answers
688 views

How to denote “powers” of a function?

I'm working with functions themselves, and I have learned that functional powers mean composition so: $f^3 = f \circ f \circ f$ But I'm looking for something that means $fff$. So $(fff)(x) = ...
1
vote
2answers
50 views

What is the definition of differentiability?

Some places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the funtion is said to be differentiable at that point. Others define it based on ...
1
vote
0answers
24 views

What function describes the frequency for each unique ratio for all possible expansions n over d where n<d?

I am hoping to solve the following problem for a scientific investigation, which relies on the probabilites of all possible expansions. What function $f(r)$ describes the frequency for each ratio for ...
1
vote
2answers
37 views

When can I not use the chain rule?

If $z=f(x,y)$ where $x=g(r,\theta), y=h(r, \theta)$, then can you give me a good reason why $$\frac{\partial^2 z}{\partial r^2} \neq \frac{\partial}{\partial x}\frac{\partial z}{\partial ...
-1
votes
1answer
31 views

Find a two variable function which is one to one but not onto

Can any of you suggest me a two variable function which is one to one but not onto? At $\mathbb{N}\times\mathbb{N} \to \mathbb{N}$ range!
1
vote
0answers
14 views

How could the rotation of an avatars head and body be determined by a single input value?

In a virtual reallity environment I want to rotate a head according to the rotation value provided by the camera sensor of a head mounted display. When the head reaches the boundary of its maximum ...
-1
votes
2answers
41 views

Integrable functions (domain)

"Verify if both of this functions are integrable in their domains: $g(x) = ( \|x\|^2 + 1)^{ -\frac{a}{2} }, \forall a>n$, domain in $ \mathbb{R} ^n$ $h(x) = \|x\|^{-\|x\|}$, defined in ...
0
votes
0answers
27 views

Is it always possible to algebraically express a function defined by a set of rules?

Let's say you have an arbitrary function defined by a set of rules such that for example: Domain $\hspace{9mm}$ Range $\hspace{5mm}$ 1 $\hspace{23mm}$ 2 $\hspace{5mm}$ 2 $\hspace{23mm}$ 2 ...
1
vote
3answers
69 views

Critical point of a function - $\Bbb R^n$ Analysis

Consider $f=(f_1,f_2,f_3): U \rightarrow \mathbb{R}^3$ a function not identically null, $f\in C^1$ and rank $3$ at every point of the open $U \subset \mathbb{R}^n$, $n \geq 3$. Show that $g(x)= ...
0
votes
0answers
34 views

What is the inverse of factorial? [duplicate]

Please write a function in any form it can be expressed in, that is the inverse function of the factorial function. By factorial function I mean $\Gamma(z)$.
2
votes
1answer
35 views

maxima and minima piecewise function

I'm exercising on maxima and minima, I think I got the point of global and local extremes but then I find this piecewise function where my teacher says that the right answer is "c". I thought the ...
0
votes
1answer
20 views

What does “decreases more slowly” mean mathematically with regard to distributions?

In a paper I'm reading, the authors state that a certain distribution "decreases more slowly than exponentially over a portion of the range". What does this mean, mathematically? Assuming $A$ is the ...
1
vote
1answer
26 views

Struggling with a problem in functions.

Suppose '$f$' is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $f(f(a))=a$ for some $a \in \mathbb{R}$ then find the number of solutions of the equation $f(x)=x$. Options given: ...
0
votes
0answers
11 views

Finding the correct asymptotic functions

in an exercise of algorithms, I have to find a function $f(n)$ that is $Ω(n^2)$ and such that for every $n>0$ is $f(n)<n^2$. I also need to find a function $g(n)$ which is $Ω(n^2)$ but not ...
0
votes
0answers
62 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
4
votes
1answer
54 views

Can someone explain to a calculus student what “dual space is the space of linear functions” mean?

I ran across this phrase today in a post and I am slightly confused. From my understanding, the dual space is the space of functions that sends a vector to a real number. There are two confusions: ...
-1
votes
0answers
51 views

History of quadratic function

For my thesis, I have to write a short article about history and importance (applications, education) of quadratic function. Could you give some papers, books, articles about it? Thanks in advance
0
votes
1answer
28 views

Stronger than contraction mapping

I'd like to know how to find a function for which $|f(a) - f(b)| < C * |a - b|^p$ where $C\in (0;1)$ (for p = 1 it's a contraction mapping, I am looking for functions for which p > 1).
1
vote
1answer
37 views

Where do the coefficient equations for Fourier series come from?

I don't see where the equations come from like: $$a_0= \frac{1}{2L} \int_{-L}^L f(x)~dx$$ And like wise for $a_n$ and $b_n$. Also where does the general formula for a Fourier series come from? If ...
0
votes
1answer
20 views

If $f_n \to f$ uniformly on compact sets, does $f_n(u) \to f(u)$ in $L^2$ and is $|f_n(u)|$ uniformly bounded?

Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$. Furthermore $f(0)=f_n(0)=0$. These functions are smooth and bijective. Also $f_n'$ is ...