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

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2
votes
2answers
40 views

Equality of functions

How a function which is not defined for some value can be equal to a function which is defined for the same value? How is $f(x) = \frac{(x-2)(x-3)}{(x-2)(x-4)}$ equal to $g(x) = \frac{(x-3)}{(x-4)}$ ...
1
vote
2answers
23 views

Number of monotonic set functions from all the subsets of some finite set to 0 or 1

Let $N=\{1,2,\ldots n\}$ be some finite set. Let $f:P(N)\rightarrow\{0,1\}$ be a function such that $A\subset B\rightarrow f(A)\leq f(B)$ I'm trying to find an upper bound to the number of such ...
1
vote
3answers
44 views

Is $g$ the unique function with this property?

Prove/Disprove: Let $A$ and $B$ be sets and let $f : A \to B$ be a function. If there is a function $g : B \to A$ such that $g\circ f = \operatorname{id}_A$, then $g$ is the unique function with this ...
0
votes
0answers
12 views

CONFIDENCE LEVEL for Median Interval

A firm wants to estimate the unknown median, m , of the height of their employees. Random Simple Size = 90 $X_{i}$ is the order statistics of the Sample Size X where height of each employee was ...
0
votes
0answers
24 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
0
votes
1answer
18 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
0
votes
1answer
32 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
1
vote
1answer
35 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
3
votes
1answer
42 views

How do I prove this function doesn't exist?

Let $g: S\rightarrow S$ be a function such that $g$ has exactly two fixed points, and $g\circ g$ has exactly four fixed points. Prove that there is no function $f:S\rightarrow S$ such that $g=f\circ ...
0
votes
1answer
30 views

Solving ODE for x instead of y

Find the general solution of the ODE. Give the largest interval over which the general solution is defined. Determine any transient terms in the general solution. $y dx - 4(x+y^6)dy = 0$ This is ...
2
votes
1answer
54 views

The “sin-cos-maximum” function

Is there some specific notation for the function $f(x):=\max\{\cos(x),\sin(x)\}$, or maybe some equivalent compact expression? Improvement: Actually, maybe a compact equivalent expression for its ...
1
vote
0answers
25 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
2
votes
2answers
46 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
-1
votes
0answers
42 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
0
votes
2answers
155 views

What method is used to find the expression of a function?

I've found some difficulties in this exercise please could you give me help? Let $f$ be a continuous function in $\mathbb R$ such that $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y)).$$ ...
-2
votes
0answers
32 views

algebra question MATH [on hold]

Find the indicated function and write its domain in interval notation. m(x) = , n(x) = x + 3, (m n)(x) = ? A) (m n)(x) = ; domain: [-5, ∞) B) (m n)(x) = (x + 3); domain: [-2, ∞) C) (m n)(x) = ...
0
votes
1answer
42 views

Find conditions on $m$ and $n$ that ensure that $f$ is a bijection.

Given: $X=\{0,1,2,...,m-1\}$, $f:X\to X$, $f(x)=nx \pmod m$. Find conditions on $m$ and $n$ that ensure that $f$ is a bijection. Progress It seems that $(m,n)=1$ but I can't prove that. I tried ...
3
votes
1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
0
votes
1answer
258 views

What does this dollar sign over arrow in function mapping mean?

In a certain function mapping like this, $x \xleftarrow{\$} \{0,1\}^k$ (Lecture Notes on Cryptography by S. Goldwasser and M. Bellare, page 18) I fail to understand what exactly does this \$ sign ...
0
votes
1answer
115 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
2
votes
2answers
84 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
0
votes
0answers
39 views

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$ [duplicate]

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function. Which of the following sets cannot be the image of $(0,1]$ under $f$? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ I think ...
2
votes
0answers
48 views

How to show that $f$ can only have at most one root in $(a,b)$ with these conditions?

Let $f: [a,b]\rightarrow\mathbb{R}$ be a differentiable function on $(a,b)$. Suppose $f$ has the following property: If for an $x \in (a,b)$, $f(x)=0$, then $f'(x)>0$. The excercise is to show, ...
0
votes
1answer
28 views

A question about sets of limit points of continuous functions.

Let $f:\Bbb{R}\to\Bbb{R}$ be a continuous function and $A\subset\Bbb{R}$ be defined by $A=\{y\in\Bbb{R}:y=\lim\limits_{n\to\infty}f(x_n)$, for some sequence $x_n\to+\infty\}$. Then $A$ is ...
3
votes
0answers
40 views

How many such functions are possible?

Let $f$ be a function from $\{1,2,3,\dots,10\}$ to $\Bbb{R}$ such that ...
-3
votes
0answers
33 views

Let $A = \{1,2,3,4\}$, and $B=\{x,y,z\}$. [on hold]

a) Give an example of a function $f: A \to B$ that is onto b) Give a function $g: B \to A$ so that $f\circ g=iB$. c) Is your function $g$ one-to-one? Justify. Any help on this homework problem I ...
0
votes
2answers
21 views

Proving that the relation from the null set to the null set is a function

How would one prove that a relation that maps the null set to the null set is a function? I tried showing that the domain of the relation is the null set, but I'm unsure of where to proceed from ...
1
vote
2answers
36 views

Order of $f(n) = 4n + 6n^3 - 8n^5$

If a function $$f(n) = 4n + 6n^3 - 8n^5$$ then the order of $f$ is: The answer I have is $\log(n)$, but I'm not sure if it's right.
4
votes
1answer
75 views

Suppose that $f: A \to B$ and $g: B \to C$ are functions.

Suppose that $f: A \to B$ and $g: B \to C$ are functions. Prove the following: (a) If $g \circ f$ is injective, then $f$ is injective. Proof. Assume that $f$ is not injective. Then ...
1
vote
4answers
28 views

Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
0
votes
1answer
35 views

$C^l$ diffeomorphism between a smooth manifold and a $C^k$ manifold

Let $M$ and $N$ be two Riemannian manifolds. $M$ is smooth while $N$ is $C^k$ manifold. Suppose there is a $C^l$ diffeomorphism between the two manifolds for $l \leq k$. Is it true that $N$ is also ...
-3
votes
0answers
53 views

Bijection from $\mathbb{Z}$ to $\mathbb{Q}$ [closed]

Can you explicitly tell me a bijection from $\mathbb{Z}$ to $\mathbb{Q}$. I need an explicit one. Thanks in advance.
0
votes
1answer
14 views

Are the roots of a smooth function, a smooth function?

Let $f(x,y)$ be a smooth function. It is given that for every $x$ there exists at least one $y$ such that $f(x,y)=0$. Is this possible to select one such $y$ for every $x$, such that the $y$'s are a ...
18
votes
2answers
474 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
3
votes
3answers
59 views

Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
3
votes
1answer
60 views

If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.

The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an ...
0
votes
2answers
53 views

$y=e^{-x}$ and $y=x$ point of intersection

How can I find the point of intersection of $y=e^{-x}$ and $y=x$ ? Here's the graph
3
votes
4answers
56 views

What is the difference between a surjective and a continuous function?

What is the difference between a surjective and a continuous function? If a function is surjective then it takes all values so it is continuous and also if a function is continuous then it takes all ...
0
votes
1answer
20 views

How can I tell if the function $f(n)=2n$ on $\mathbb Z$ is one-to-one, onto, or both?

The domain of the function is the set of all integers. The codomain of each function is also the set of all integers. $$f(n) = 2n $$ I was thinking that the function is one-to-one but I don't know ...
17
votes
3answers
708 views

What kind of “mathematical object” are limits?

When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
-2
votes
1answer
14 views

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$ [closed]

Value of $K$ so that $f(x)=3x^3+6x^2+KX-4$ has the same remainder when it is divided by $x-1$ and $x+2$. Thanks for the help!
1
vote
1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
4
votes
1answer
91 views

Condition on vector-valued function

Does anyone have any ideas on how to show that the following is true: Let $\Omega \subset \mathbb{R}^{n}$ be open and bounded. Consider vector-valued function $$f: \Omega \times \mathbb{R} \times ...
0
votes
1answer
21 views

Relaxing Monotonicity of a Function $f:\mathbb{Z}\rightarrow \mathbb{R}$

Suppose a function $f:\mathbb{Z_+}\rightarrow \mathbb{R}$ fails monotonicity, but not by much. For example $f(2)= .3$ and $f(z)=1/z$ otherwise. Here there exists a single point where the function is ...
0
votes
1answer
25 views

Finding the domain of a fourth-root of an equation with a term to the fourth

Hi – I've got this question about finding the domain of a function, and I got the answer, but the method I used is quite different from the explanation provided. My question is: Is my method flawed in ...
3
votes
2answers
30 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
0
votes
1answer
10 views

how to prove that a given function is univalent

I have to prove that following function is univalent $f(z) = z^2 +3z +1, ~|z|<1$ in complex plane. What I tried is: Let $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
0
votes
1answer
54 views

Find $\int_2^{2.2}f(x)\,\mathrm dx$ given $f(x)=x^4-3x^3+9x^2+22x+6$.

$f(x)=x^4-3x^3+9x^2+22x+6$. Find $\int_2^{2.2}f(x)dx$ by finding $f(x-2)$ This is in a non-calculator paper which is why $f(x-2)$ is meant to be obtained (it's supposed to made the maths possible to ...
2
votes
0answers
17 views

Proving a basic result about Holder continuous functions

Let $V$ be a open convex set. We will say that a function $m$ has the order of smoothness $p$ on $V$ with $p=l+\gamma$, where $l \geq0$ is an integer and $0<\gamma\leq1$ and will write $m\in ...
1
vote
2answers
48 views

Bijection between $\mathbb R^2$ and $(0,1)$ [closed]

I want to prove the sets have same cardinality: $\mathbb R^2$ and $(0,1)$ Please Help.