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

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2
votes
2answers
22 views

Then the value of $ [f(2)] $ where [.] represents the greatest integer function is?

A differentiable function f is satisfying the relation $$f(x+y) = f(x) + f(y) + 2xy(x+y) - \dfrac{1}{3} $$ $ \forall $ $ x , y $ belongs to $\Re$ and $$lim_{h \to 0} \dfrac{3f(h)-1}{6h} = ...
1
vote
2answers
32 views

bound of integrable function

I want to prove the following conjecture: if an integrable function $f(x)$ is continuous on (0,T] and unbounded at $x=0$, then there exists positive $M$ and $\alpha\in(0,1]$ such that $$ |f(x)|\leq ...
3
votes
4answers
46 views

Show that $f$ is a decreasing function

It's given that $f(x)=\frac{1}{x^3}-x^3$ for $x>0$ show that $f$ is a decreasing function. My attempt $f'(x)=-3x^{-4}-3x^2$ $x^6=-1$ How to continue by my attempt ?
1
vote
2answers
35 views

Proof of onto and one-to-one functions, composition

I want to prove this: Let $f: A \to B$ and $g: B \to C $ be functions. if $g \circ f$ is onto, and $g$ is one-to-one, then f is onto. Here is what I have done, can someone please verify my work: ...
1
vote
0answers
8 views

Indexing interactions between and withing entities

I'm trying to create/find an index to compare/order systems with multiple entities based on the diversity of the interaction between the entities. Assume you have few systems of entities that can ...
0
votes
1answer
21 views

Predictability of what $\lfloor n\log n\rfloor$ “skips”

TL;DR: is there any way to tell what numbers will not be present in the function $\lfloor n\log n\rfloor$ under a given upperbound? I am writing a program that will calculate the sum of the gaps in ...
2
votes
3answers
244 views

Showing surjectivity of a certain function [closed]

Let $X=\Bbb R/{\sim}$, where $\sim$ is the equivalence relation that $y-x = 2n\pi$. Consider the function: $f: X \rightarrow S^1$ ; $f(x) = (\cos (x), \sin (x))$. Show that this function is ...
2
votes
1answer
47 views

Prove using algebra of continuous functions that $f$ is continuous in $\mathbb{R}$

Let us consider $f : \mathbb{R} \to \mathbb{R}$ defined by $$f(x) =\begin{cases} x^2 \sin \frac{1}{x},& x \neq 0\\ 0,& x = 0\end{cases}.$$ By using algebra of continuous functions ...
0
votes
3answers
20 views

Finding functional values

I am dealing with a certain function f satisfying f(3x) = 3f(x) for any positive real number x. Also f(x) = 1 - |x-2| for all real x in the interval [1,3]. Now I am trying to formulate a general ...
1
vote
1answer
32 views

Sets,transversals,PT property,cardinals

A transversal of a family $S$ of sets is an injective choice function. $PT(\lambda,\chi)$ means, if $S$ is a family of $\lambda$ sets,each of cardinality $<\chi$,and every subfamily with ...
3
votes
0answers
64 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
3
votes
1answer
38 views

Injective function on the domain of natural numbers

Find all injective functions $f:N \rightarrow N$ such that $$f(f(m)+f(n))=f(f(m))+f(n)$$ Where $m,n$ are natural numbers.
3
votes
0answers
23 views

an integral equation in a function with two arguments

Say we are given $C(s,t)=\min(s,t)+\zeta st$. How can we solve $$g(s,t)=C(s,t)+\lambda \int_0^1g(s,u)C(u,t)du.$$ Looking into some text books on integral equations I see that most of the kernels, $C$, ...
7
votes
3answers
60 views

At least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$

Let $a\in(0,1),f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3$,$g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $. Prove that: For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$ since ...
0
votes
1answer
46 views

Need help to understand Uniqueness of Lifts theorem's proof.

Theorem: Let $p:E \to B$ be a covering map. Fix $b_0 \in B$ and $e_0 \in p^{-1}(b_0)$. Let $f: X \to B$ be a continuous map with $f(x_0)=b_0$ and $X$ is connected. Suppose $g_1,g_2: X \to E$ are ...
0
votes
0answers
11 views

Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
2
votes
1answer
37 views

What is the least possible value of $𝑓(999)?$ [duplicate]

Let $𝑓$ be a one-to-one function from the set of natural numbers to itself such that $𝑓(𝑚𝑛) = 𝑓(𝑚)𝑓(𝑛)$ for all natural numbers $𝑚$ and $𝑛.$ What is the least possible value of $𝑓(999)?$ ...
1
vote
1answer
42 views

Discrete Mathematics Relations and Functions

I'm stuck on this question, and I'm unsure if my though process for the question is correct or not, as my understanding of relations isn't the greatest. Let A = {1, 2, 3, ..., n} where n is a ...
1
vote
1answer
32 views

Define the function $f:(0,1)\to (0,1)$

Define the function $f:(0,1)\to (0,1)$ by $$ \displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} ...
1
vote
1answer
13 views

Conditions for a smooth optimizer?

Consider a function $f:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}$. I am trying to determine conditions (on $f$ and/or $X$) under which the maximizer defined by \begin{align} \hat x(\alpha) = ...
1
vote
4answers
303 views

What a good approach will be to solve this problem?

I know that this function from A to A is 1-1 and also onto. How many functions like this exists ? The set A contain 12 elements. $$\forall a \in A $$ $$f(f(a)) \ne a{\rm{ , }}$$ $$f(f(f(a))) = ...
1
vote
3answers
47 views

Why can't this mixed function be inverted?

Given the function $$y=Ax + B\sqrt x$$ where $A$ and $B$ are real constants, $x$ is real and $x > 0$ I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible ...
-1
votes
2answers
52 views

How do I find lowest upper bound and greatest lower bound when dealing with functions? [closed]

Here is my problem: I have to find the integer that is the highest lower bound for the roots of $$f(x)=x^4-3x^2+2x-4$$ I am not sure how to do this and the book I am using does not explain it very ...
0
votes
1answer
24 views

Find a surjective Function f

I'm trying to find a surjective function f: $\mathbb N^\mathbb Z \rightarrow \mathbb N \times \mathbb Z$ where $\mathbb N^ \mathbb Z $ is the set of all of all functions $\mathbb Z \rightarrow \mathbb ...
-3
votes
1answer
19 views

Scale Proportionally a Polynomial [closed]

I have 3 polynomial functions that all overlap (to form a shape): I need to scale the polynomials down to half the current size. How do I do this? It must be scaled down proportionally ($x$ and ...
1
vote
1answer
30 views

If $g(x)=2f(x/2)+f(2-x)$ and $f''(x)<0$ for all lying in $(0,2)$ how to find the interval where $g(x)$ increases?

If $$g(x)=2f(x/2)+f(2-x)$$ and $\hspace{.1cm} f''(x)<0$ for all lying in $(0,2)$ how to find the interval where $g(x)$ increases? I differentiated it once and twice but I'm not being able to draw ...
0
votes
0answers
30 views

Period of g(x):functional equation based problem $f(x,y)=f(2x+2y,2y-2x)$

Consider a real valued function $f$ which satisfies $f(x,y)=f(2x+2y,2y-2x).$ There is another real valued function $g(x)=f(2^x,0)$, how do I find the period of $g(x)$? Note:f is not constant ...
1
vote
0answers
42 views

Is there a name or symbol for continuous and bounded functions?

For example, I have a function $f$ that is continuous and satisfies $|f| \leq M \lt \infty$ Is there a succinct notation to decribe this function without having to specify continuity and boundedness ...
1
vote
1answer
43 views

Let $f(x)=(x-a)g(x)$. Calculate $f '(a)= g(a)$.

Assuming that $g$ is continuos. What is $f(a)$ in this case? Can you give an explanation for this proof? How can it be possible to ask about a function of another variable of an existing function? Can ...
3
votes
3answers
124 views

How many functions $f: A\rightarrow A$ exist without (?) any $f(x)=x$

The definition: $\mathcal A = \{1\cdots12\}$ $\mathcal f: A \rightarrow A$ for each $\mathcal x \in A$, $\mathcal f$ is defined $\mathcal f(f(x)) \neq x$ and $\mathcal f(f(f(x))) = x$ How many ...
0
votes
1answer
26 views

composite function

A colony of rabbits in outback Australia was studied. The colony began with 10 rabbits (5 pairs). Each female produced an average of 7 offspring of which 3 were female. 50 days later the original 5 ...
0
votes
1answer
36 views

Determine function even or odd. Why? [closed]

I've been asked to determine whether a function is even or odd. Why is this useful?
2
votes
2answers
41 views

How to tell if a function has rotational symmetry? [closed]

How to spot rotational symmetry on(or in, of?) a function? If I have the function $f(x)={{5a^2+6ax+9x^2}\over {a+3x}}$, how can I know it has rotational symmetry about the point $(-{a\over 3},0)$? Is ...
1
vote
2answers
46 views

Simplification of $n^{1/\sqrt{\log n}}$

I would like to simplify this function, how can we do it ?
1
vote
2answers
25 views

Getting the quadratic function given the vertex and one point.

Find out the quadratic function for the parable that contains the point $(1,1)$ and the vertex $(-2,3)$. The notes I got are pretty vague: $$b = 4\frac{-2}{9} = \frac{-8}{9}\\ c = ...
0
votes
0answers
52 views

Standard notation for the transform that turns a function $A \rightarrow (B \rightarrow C)$ into a function $B \rightarrow (A \rightarrow C).$

Suppose we're given sets $A,B$ and $C$. Then to each function $f : A \rightarrow (B \rightarrow C)$, we can assign another function $F : B \rightarrow (A \rightarrow C)$ by defining: $$F(b)(a) = ...
7
votes
4answers
791 views

What do we call a “function” which is not defined on part of its domain?

Before the immediate responses come in, I realize that a properly defined function means that it is defined for every value in its domain. My question is this: if $f:A\to B$ has the property ...
1
vote
1answer
47 views

Proving $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ function is bijective.

Given $f: \mathbb{R}\rightarrow\mathbb{R}$ is a $C^1$ function and $\forall t.|f'(t)| \leq c < 1$,$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is defined as $g(x,y) = (x + f(y), y+f(x)).$ I am trying ...
0
votes
1answer
24 views

Ease-in function

I need a function for ease-in effect, exactly like this ease-in-out function, but without the end (infinite). I've tried two variants that I don't like too much for a while. The first one never ...
1
vote
1answer
14 views

How do I show that f(x) is independent for the variationproblem?

The variationproblem where f(x) is a reel differential function defined in the interval [0,1].
-2
votes
1answer
76 views

What is the difference between an function and functional?

Can someone give an example that would point out the difference between a function and a functional in a very simple way?
2
votes
4answers
83 views

If $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function?

Like the title reads, if $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function? Assume $x,y,\alpha\in\mathbb{R}$.
0
votes
0answers
14 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
1
vote
1answer
59 views

Function for this series of numbers

I have $f(n)$: $f\left(1\right)=1$ $f\left(2\right)=2$ $f\left(3\right)=3$ $f\left(4\right)=3$ How can I solve the next $f(n)$? I came to this problem while been working on my algorithm. Here is ...
0
votes
3answers
53 views

The range of the function $f(x)=\frac{\sin x}{\sqrt{1+\tan^2x}}-\frac{\cos x}{\sqrt{1+\cot^2x}}$

Find the range of the function $f(x)=\frac{\sin x}{\sqrt{1+\tan^2x}}-\frac{\cos x}{\sqrt{1+\cot^2x}}$. By simply looking at the problem and simplifying trigonometrically,it looks as if range is zero ...
8
votes
1answer
85 views

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$ $(A)\frac{1}{10}\hspace{1 cm}(B)10\hspace{1 cm}(C)1\hspace{1 cm}(D)$cant be obtained I could not solve this question.I ...
0
votes
1answer
35 views

If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$…

Question: If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$, then $f(f(x))+f(f(\frac{1}{x})) = ?$ (a)p (b)2p (c)-p (d)-2p Attempt: ...
1
vote
3answers
24 views

Basic question about showing bijection when a function is defined piece wise

Hello there everyone, I was told that the standard proof for showing that $$| \mathbb{N} | = | \mathbb{Z} |$$ is to define $$f: \mathbb{Z} \to \mathbb{N}$$ as $$f(x)= \begin{cases} 2x &\text{ ...
-1
votes
1answer
59 views

How to sketch a function which is 2 to the power of x [closed]

Hi I'm looking for help to sketch a function. $$ (e)\quad Sketch\quad the\quad graphs\quad of\quad each\quad of\quad the\quad following\quad functions:\\ (i)\quad f:\mathbb{R}\rightarrow ...
0
votes
1answer
29 views

Which functions can be represented as matrices?

I was reading the intuition for the associativity of matrix multiplication and it was given to be analogical to composition of functions. So which functions can be represented as matrices and how?