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

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0
votes
3answers
26 views

Limit of a function w

If $f(x, y)$ is a continus function, defined in whole $\mathbb R^2$, then the limit $$\lim_{(x, y)\rightarrow(2,2)}f(x, y)(x-1)(y-2) $$ The solution is $0$, but how? A very elaborative explanation ...
4
votes
2answers
38 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
4
votes
6answers
66 views

Domain of $\frac{1}{\frac{1}{x}}$

Let $f(x)=\frac{1}{x}$, then we have $f^{-1}(x)=\frac{1}{x}$. So $f(f^{-1})=\frac{1}{\frac{1}{x}}=x$. My question is, what is the domain of $f(f^{-1})(x)$? is it everything? or everything but zero? ...
1
vote
3answers
19 views

Intersection of inverse images

Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$, $$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective. But why for the inverse, suppose that $E$ and ...
0
votes
1answer
15 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
0
votes
0answers
15 views

Questions about functions, their domains and codomains.

I am playing around with equations about functions in general and have some questions. Question 1 If I have some functions $f,g\colon X^2 \rightarrow Y$ such that $f(x,y) = g(x,c)/g(y,c)$ then can ...
0
votes
1answer
12 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
0
votes
1answer
39 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
0
votes
1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
-3
votes
1answer
25 views

Arccos function [closed]

How to look for condition (limits of x): Or, how to get domain limits for $-1 < = \frac{( x-1)} {(2 x +7) } < = 1 $ for using in -1 <= argument <= 1 ( to get real values of ...
1
vote
1answer
24 views

Inequality from a property of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. How can I prove that for each $x$, there is $c$ such that $f(x)+c(y-x)\leq f(y)$ for all $y$? One of the difficulties to solve is $f$ does not ...
0
votes
0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
0
votes
1answer
27 views

Empty function, what is it?

I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?
0
votes
0answers
9 views

general inverse of multivariable function

Given $m<n$. Let $f:N\subset\mathbb R^m\mapsto \mathbb R^n$ be a differentiable function. I am looking for the condition(s) such that we can find a function $g:im(f(N))\mapsto \mathbb R^m$ ...
1
vote
1answer
35 views

Who is growing faster?

I am trying to prove that $\lim_{n\to \infty} { 2^{n^2} \over n!} = \infty$. I can't use l'Hôpital's rule (or I dnon't know how) and I don't recall any other method which could help me. It also isn't ...
0
votes
2answers
28 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
1
vote
1answer
11 views

Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
0
votes
1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
1
vote
4answers
58 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
-3
votes
0answers
17 views

Is these functions periodic ones? [closed]

Determine whether the function is periodic. If it is periodic, find the smallest (fundamental ) period $f(x)=cos(2x) + 3sin(\pi x)$ $f(x)=sin(2x)-cos(5x)$ thanks in advance
0
votes
0answers
39 views

Rewriting a function to speed up computation

the purpose of this question is to find the fastest way to calculate a function on a (x,y,z) grid using a programming language. It is more related to maths however, as it concerns recasting a function ...
4
votes
2answers
131 views

What is the opposite category of $Set$?

In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there ...
-1
votes
1answer
11 views

Number of different functions

Suppose that $A$ has exactly $m$ elements and $B$ has exactly $n$ elements. How many different functions are there from $A$ to $B$? The answer is given by $n^m$ but i don't know how to get that And ...
1
vote
2answers
32 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
2
votes
2answers
24 views

Does there exist a bijective mapping of an open interval with the corresponding closed interval having only finitely many points of discontinuity?

Given $a<b$, is there a bijection $f \colon [a,b] \rightarrow (a,b)$ such that $f$ be continuous except at finitely many points only? I know that there does exist a bijection of $[a,b]$ with ...
0
votes
0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
0
votes
2answers
31 views

Find $\lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1}$ and $\lim_{x\to-\infty} \frac{e^{2x}-1}{e^{2x}+1}$

How do I calculate $\displaystyle \lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1} \ , \ \lim_{x\to-\infty}\frac{e^{2x}-1}{e^{2x}+1}$. Please help me. Thanks!
1
vote
0answers
19 views

Is a sine wave plus the sum of its odd harmonics symmetrical around the x axis at half the period of the fundamental?

I have a function such that $$x(t)=A_1 \sin(2 \pi f t+\phi_1)+A_2 \sin(2 \pi (3f) t+\phi_2)+...+ A_n \sin(2 \pi ((2n+1)f) t+\phi_n)$$ Is such a function symmetric around the point that is half ...
-4
votes
1answer
36 views

Riemann hypothesis: An query about the primes [closed]

How can we describe the Riemann hypothesis easily? What is the connection between the Riemann hypothesis and prime numbers?
1
vote
0answers
68 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
28 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
62 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
147 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
71 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
24 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
42 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: ...
1
vote
0answers
73 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
40 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 ...