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

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-4
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1answer
33 views

Riemann hypothesis: An query about the primes [on hold]

How can we describe the Riemann hypothesis easily? What is the connection between the Riemann hypothesis and prime numbers?
1
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0answers
66 views

Follow-up regarding $f(x,y)\cdot f(y,z) = f(x,z)$

I posted this question about a year ago. And I started thinking about it again. The question as follows: Under what conditions does $f(x,y) \cdot f(y,z) = f(x,z)$ not imply that $f(x,y)$ can be ...
1
vote
1answer
34 views

Is there a basis for the continuous functions space?

I've been searching all over the Internet for this but without finding a satisfying answer. This might be a dumb question, but I would like to know the answer anyway. Is there a set of continuous ...
0
votes
0answers
24 views

Can the image of this specific function sit on a variety?

Let $d_x<d_y$, $X \in \mathbb{R}^{d_x}$ and $Y=f(X) \in \mathbb{R}^{d_y}$ defined implicitly by the equation: $$ A X +B \log Y + C \log (1-Y) =0$$ where $A \ (d_y \times d_x)$, $B \ (d_y \times ...
1
vote
1answer
56 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
0
votes
1answer
15 views

Map Distributing (Backwards) Through a Composition

I'm reading through "An Introduction to The Theory of Lists" and am having a hard time figuring out how to prove: $$ (f \circ g)* = (f*) \circ (g*) $$ found on page 5, where the asterisk (*) stands ...
4
votes
1answer
146 views

Is the range of an injective function dense somewhere?

Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? ...
0
votes
1answer
8 views

Prove the uniqueness of functional equation

How do one prove the uniqueness of a functional equation. (elementary) not functional analysis class... For example, if we have $f(x+y)=x+f(y)$ and $f(0)=1$. Letting y=0, we obtain $f(x)=x+1.$ But ...
0
votes
1answer
32 views

Find out $f(n)$ where n is an integer

Suppose I need a function $f(n)$ such that $f(n)$ is odd when $n=4,12,20,28...$ and even when $n=8,16,24,32...$. Then the answer would be $f(n)=\frac{n}{4}$. Similarly, now suppose I need $f(n)$ such ...
1
vote
1answer
25 views

Finding absolute extrema

The function is $f(x,y)=\sin x+\cos y+\sin (xy)$, which on {$(x,y)|0\le x\le 2\pi,~0\le y \le 2\pi$} I want to find the absolute extrema of function $f(x,y)$. I try to find gradient of function $f$, ...
1
vote
0answers
26 views

exact angle between the functions $p(x)=3x-4$ and $q(x)=9x-5$ over $0\leq x\leq 1$

I was attempting a question, which gave the formula for the angle theta between two functions $f(x)$ and $g(x)$ over $a\leq x\leq b$ (note the question defined the meaning of norm and inner product as ...
-1
votes
1answer
20 views

a simple question about compositions of functions

If $f(v) = \frac{1}{v^2 + 5}$ and $g(v) = \sqrt{v+4}$, how do you find $g \circ f$? Please provide steps.
3
votes
1answer
68 views

Continuity of $a^x$ when it's defined by the ordinary way

I've searched for the discussion of proving the continuity of exponential function, in most cases the function is defined by power series or inverse of log function where the log is defined by ...
1
vote
2answers
32 views

figure out $f(n)$ under given conditions

Suppose I need a function $f(n)$ such that $f(n)$ is odd when $n=4,12,20,28...$ and even when $n=8,16,24,32...$. Then the answer would be $f(n)=\frac{n}{4}$. Similarly, now suppose I need $f(n)$ such ...
0
votes
0answers
22 views

In need of formula: Gravity at Specific Coordinates [closed]

Doing Research on the gravitational pull at a specific set of coordinates. Does anyone know how to solve this mathematically? Please Help. Thanks
0
votes
1answer
41 views

Function in Polar Coordinates

Let $f,g:I\to\mathbb{R}$ be two function in $C^{k}(I)$, with the property that $f^2(t)+g^2(t)=1, \ \forall\ t\in I$. Is there a function $\theta: I\to\mathbb{R}$, $\theta\in C^{k}(I)$, such that: ...
0
votes
0answers
60 views
+50

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
0
votes
1answer
15 views

How find this value if such $f[f(x,y),z]=f(z,xy)+z,x,y,z\in R$

Question: let function $f(x,y)$ such $$f(x,0)=1,x\in R$$ (2):$$f[f(x,y),z]=f(z,xy)+z,x,y,z\in R$$ Find the value $$f(2014,6)+f(2016,6)=$$ My idea: let $y=0$, then $$f(1,z)=f(z,0)+z=1+z$$ I ...
0
votes
2answers
18 views

Functions where a composite gets $\mbox{id}_A$ but not $\mbox{id}_B$ and another function $\mbox{id}_B$ and not $\mbox{id}_A$

this question is really causing me to pull my hair out. I have to find a function $f : A \to B$ such that all of the three conditions are true for the same function $f$: (1) there is a function $f_1 ...
1
vote
0answers
37 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
0
votes
1answer
21 views

$g(f(n))\in o(g(n)/n)$ for any $f(n)\in o(n)$

Let $f:\mathbb{N}\rightarrow\mathbb{R}$ be a function such that $f(n)\in o(n)$. Is it always possible to find a function $g:\mathbb{N}\rightarrow\mathbb{R}$ such that $g(f(n))\in o(g(n)/n)$? I'm ...
0
votes
2answers
24 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
1
vote
1answer
13 views

$g(n)\in\omega(n^r)$ but $g(f(n))\in o(n^{r-1})$

Let $f:\mathbb{N}\rightarrow\mathbb{R}$ be a function such that $f(n)\in o(n)$. Is it always possible to find a function $g:\mathbb{N}\rightarrow\mathbb{R}$ and a constant $r>1$ such that ...
0
votes
1answer
21 views

Making a function for unbounded variables that is bounded

Ok here is what I need I want to build a formula to implement into a ranking system on my website its been like 20 years since I was in school so help me out I need like 6 variables that equal at ...
0
votes
1answer
30 views

Function to center text

I am remaking 2048 in java. I am trying to center the text of the number inside the box with a function, but I have some trouble translating the text since the $x$ and $y$ are on the bottom left of ...
1
vote
1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
4
votes
3answers
783 views

Limit of a 0/0 function

Let's say we have a function, for example, $$ f(x) = \frac{x-1}{x^2+2x-3}, $$ and we want to now what is $$ \lim_{x \to 1} f(x). $$ The result is $\frac{1}{4}$. So there exists a limit as $x \to 1$. ...
-3
votes
1answer
33 views

Help with calculus 12 question [on hold]

So I am having trouble with my calculus homework! The question is: Find a value for $a$ so that the function $$f(x)=\begin{cases}x^2-1& x<3\\2ax& x\ge 3\end{cases}$$ is continuos. ...
0
votes
0answers
19 views

How to find the period of a sinus function from graphical representation

I have a graphic showing a sinus function. The x and y axis are real numbers. How can i express the period of the sinus in radians ? Thanks edit 1 : Ok ok "sine" "sinus" is from french ^^ I know ...
0
votes
1answer
28 views

How do I plot the following function? [closed]

I need to plot the following function, in order to get the intuition for optimization theory: $$ f(x) = \begin{cases} x , & \text{if $y=x^2$} \\ 0 , & \text{otherwise} \end{cases} $$ How do ...
2
votes
1answer
53 views

Special feature of the function f(z) = $|i + z|^2 + az + 3$

I have to solve following problem: Find all the values of a (a is a real number) that the function f : $f(z) = |i + z|^2 + az + 3$ (z is a complex number, i is an imaginary unit) has a following ...
1
vote
1answer
30 views

True or False, limit, functions questions. Does limit exist?

True or False Let a be a real number, and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. a) For each natural n, the function ...
-1
votes
2answers
47 views

What is rule of this function?

I have these values.these are inputs and outputs of a function.I want to find rule of function.input is N. ...
0
votes
1answer
28 views

Solving Polynomial Equations and Inequalities

The distance, in km, of a ship from its harbour is modeled by the function $d(t)= -3t^3 + 3t^2 + 18t$ where $t$ is the time elapsed in hours since departure from the harbour. a) When does ...
-1
votes
3answers
32 views

Question related to functions [on hold]

Suppose $f(x+y)=f(xy)$ for all $x,y\in\mathbb{R}$. Then how to prove that $f$ is a constant function? Please solve using simple mathematics and no calculus.
0
votes
3answers
54 views

Showing that two vector spaces aren't isomorphic?

Here is a part of an exercise (from a book) I can't figure out how to solve : Les $V$ be the set of all functions $f: \mathbb{N} \to \mathbb{R}$. We define also the functions $e_i(n)$ by $e_i(n)=1$ ...
0
votes
1answer
12 views

Does a constant of integration changes the shape of a distribution?

Let $f(x)$ be the frequency distribution of the variable $x$. Let assume that $\int^{\infty}_{-\infty} f(x) ≠ 1$. Let $g(x) = C f(x)$ such as $C$ is the constant of integration so that ...
1
vote
2answers
71 views

Linear functionals and dual bases

How do I tackle this question? I am a little hazy on linear functionals and integral signs.
1
vote
1answer
35 views

Problem with itinerary of a coding problem with infinite 1's

If $f(x)=2x \ mod \ 1$ on $[0,1)$. Then if we code $x \in [0,1)$ with its itinerary w.r.t. the partition $P_0=[0,1/2)$ and $P_1=[1/2,1)$. Can you show that there is no point $x$ whose itinerary has ...
2
votes
0answers
28 views

Finding the extrema of $f(a):=\left\vert\frac{4}{3a}\sin\left(\frac{a}{4}\right)\left(4-\cos\left(\frac{a}{4}\right)\right)-1\right\vert$

Let $0<a<\pi$. Find the minimum and maximum: $$f(a):=\left\vert\frac{4}{3a}\sin\left(\frac{a}{4}\right)\left(4-\cos\left(\frac{a}{4}\right)\right)-1\right\vert.$$
0
votes
1answer
45 views

Show that the $\lim_{n\to+∞} 1/\sqrt{n} = 0$.

Show that the $\lim_{n\to+∞} 1/\sqrt{n} = 0$ So basically my assistant told me i have to show first of all that x→√x is a strictly increasing function. So i did $√x_n < √x_{n+1}$ so $√x_n - ...
0
votes
1answer
36 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
0
votes
1answer
36 views

Intermediate Value Theorom

Show that there is at least one negative solution to $e^{x} = -x$ I understand that the intermediate value theorem takes two points to check if the continuous line crosses a section. How would I ...
0
votes
1answer
30 views

Simplifying the difference quotient $\frac{(x + h)^3 - x^3}{h}$.

For the function $f(x) = x^3$, I have the difference quotient: $$ \frac{(x + h)^3 - x^3}{h} $$ I tried changing the $(x + h)^3$ to $(x + h)(x^2 - xh + h^2)$ that I know to see if I could get ...
0
votes
4answers
35 views

Can someone walk me through how this expression simplifies to y/x?

I am just wondering how this equation comes to be: it is from an economics problem involving marginal utilities. I have my two variables, $x$ and $y$. Intuitively, how does $$\frac{0.5\times ...
0
votes
1answer
11 views

Proving one-to-one with 2 variables

$f: Z×Z -> Z$ where $f(m,n) = 2n -m$ In my class we are taught to solve this by simplifying: $2n_1 - m_1 = 2n_2 - m_2 $ However, I don't understand how this proves or disproves that $f(m,n) = ...
1
vote
2answers
19 views

Describe the preimage of the set

I've been stumped on this problem for a while now, unable to find many resources to help me understand how to describe a preimage of a set given a function like this one. Let $f$: $\mathbb Z \to ...
2
votes
3answers
56 views

l'Hôpital and it's use in derivation

In for example $$\lim_{x\rightarrow 0} \frac{e^{ax} - 1 - ax}{1 - \cos x}$$ We would use l'Hôpital rule and derive it twice to get $a^2$ How do you see this when just looking at the given function, ...
2
votes
0answers
23 views

Integers and funtional equation [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
0
votes
2answers
42 views

Showing there is no invertible function $f: \mathbb{R} \to \mathbb{R}$

I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof. The easiest would be ...