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

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0
votes
1answer
73 views

Is the derivative in $C(\overline{\Omega})$?

Let $\Omega\subset\mathbb{R}^2$ a bounded domain and consider $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I would like to know if then $$ 1+\frac{\partial u}{\partial x}\in ...
0
votes
3answers
60 views

Calculate the limit:

I need to calculate: $$\lim_{x\rightarrow \frac{\pi }{4}}\frac{\sin2x-\cos^{2}2x-1}{\cos^{2}2x+2\cos^{2}x-1}$$ I replaced $2\cos^{2}x-1=\cos2x$ and $\cos^{2}2x=1-\sin^{2}2x$, so this limit equals ...
2
votes
2answers
60 views

Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general

Let $f:X\to Y$ be a function and $E\subseteq X$ and $F\subseteq X$. Show that in general $f(E − F)\nsubseteq f(E) − f(F)$. I have no idea about how to prove this; and could anyone please explain ...
2
votes
3answers
161 views

One to one and bijection in $\mathbb{Z}^2$

I have the following: $f(m,n) = (3m+7n, 2m+5n)$ and I want to know if it is a bijection and if so, fine the inverse as well. Here's my approach: Suppose $f(m_1,n_1)=f(m_2,n_2)$ then: $$ ...
4
votes
2answers
569 views

Commutativity of iterated limits

The following is a weird result I've obtained with iterated limits. There must be a flaw somewhere in someone's reasoning but I can't discover what it is. The problem is that, in general, iterated ...
1
vote
1answer
45 views

At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
1
vote
1answer
65 views

Find all functions over naturals that they hold an equality

The task itself is not that hard i'd say. I have to find all functions $f: \mathbb N\rightarrow \mathbb N$ that equality $f(\pi(n)) = \pi(f(n))$ is true. Where $\pi(n)$ stands for ANY permutation over ...
4
votes
1answer
182 views

How can be a set of partial isomorphisms defined from a n-back-and-forth system?

While studying partial Ebbinghaus-Flum's Mathematical Logic, I came across the partial isomorphism definition, as build upon an $n$-back-and-forth system. Consequently, the question I raise in the ...
0
votes
1answer
41 views

Inverse Transformation

Consider the coordinate transformation $$ \varphi\colon\mathbb{R}^2\to\mathbb{R}^2, (x,y)\mapsto (y-\arctan(x),y+\arctan(x)). $$ To make it more easy, I set: $$ ...
2
votes
2answers
1k views

Find intersection points of two functions

I have $f(x)=\sqrt{3x}+1$ $g(x)=x+1$ My thinking was that at the intersection points both will be equal to each other so $\sqrt{3x}+1=x+1$ $\sqrt{3x}=x$ However I don't know where to go from ...
0
votes
1answer
74 views

Increasing function by derivative?

I would like to show the following function is increasing for $x \geq 1$ $$ x \frac{(b+x)^k - (a+x)^k}{(b+x)^k - a^k} $$ where $b > a > 0$ and integer $k \geq 1$. It is easy to prove by ...
0
votes
2answers
55 views

Are complex conjugates unique?

I'm trying to decide if a function $\varphi:\mathbb{C}\rightarrow \mathbb{C}$ defined by $\varphi(\alpha)=\bar{\alpha}$ is onto or one to one. I know it will be onto because every element in ...
1
vote
0answers
45 views

Text Generating Functions: Do they exist?

This is a little far out question, but just curious: is it even possible to have a non-high-degree-polynomial function (as in polynomial regression function) that could generate a sentence of say, 10 ...
3
votes
3answers
52 views

Logarithmic function

Compute the value of $f\bigl(\frac{1}{400}\bigr)$ if the function is defined as follows : $f(xy) = f(x) + f(y)$ and $f(4)= 16$
1
vote
1answer
165 views

When to rationalize numerator and/or denominator?

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
1
vote
3answers
428 views

How to evaluate this limit of irrational function?

$$\lim_{h \rightarrow 0}\frac{5}{\sqrt{5h+1}+1}$$ Some things I'm confused about: 1) Why should we rationalize the denominator? It won't get rid of the square root, it will just move it to the ...
4
votes
2answers
180 views

What is a polynomial and how is it different from a function?

I have a problem that asks me to find a polynomial $P(x)$ so that $P(3)$ is 9. Now I can say with certainty that $P(x)$ can be $x^2$. This is a second degree polynomial. But what about functions ...
0
votes
1answer
58 views

Need Domain and Function Question Help [closed]

I have have the hardest time when it comes to functions could someone give me some tips on how to complete this story problem thanks. Question: You and three friends sign up for a golf tournament. ...
0
votes
2answers
93 views

What function can achieve the following?

So I have some values which are computed linearly. But I want to stress the middle range more so I want the values to be "transformed" into something like this: So basically, say for $x$ between ...
1
vote
3answers
126 views

Compute integral on the interval 0,1

Sorry but I have a problem I am student in first year in economics. I don't have enough knowledge on integration. I want to compute this integral, for $a>2$, $a>i>0$, $b\in R$ and $d\in R$. ...
1
vote
4answers
99 views

How can I determine this function?

If i know that$$ f(3)-f(-1/2) = 7$$ and $$ f(2)-f(1/2) = 3 $$ and $$f(3/2)-f(-2) =7$$ How can I determine the function?
0
votes
1answer
75 views

Existence of an injection [duplicate]

Let $A$ and $B$ be two sets. Prove the existence of an injection from $A$ to $B$ or an injection from $B$ to $A$. I don't know how to proceed, since I don't have any information on $A$ or $B$ to ...
1
vote
8answers
257 views

Prove $f(x)= e^{2x} + x^5 + 1$ is one to one

Prove $$f(x)= e^{2x} + x^5 + 1$$ is one to one. So my solution is: Suppose $$ f(x_1)=f(x_2),$$ then I am stuck here: $$e^{2x_1}-e^{2x_2}=x^5_1 -x^5_2.$$ How do I proceed? Also after that I ...
0
votes
1answer
35 views

Limit of functions

I need to give a counterexample to the following statement: If $ \lim_{x \to 0} \left( \frac {f(x)}{g(x)} \right) = 1 $, then $ \lim_{x \to 0} \left( f(x) - g(x) \right) = 0 $. The problem is I ...
1
vote
0answers
38 views

Determining equations for smooth quadratic-linear function.

I am trying to find a smooth function that goes from quadratic to linear, with the linear segment being of a given length $\ell$. The only known values are $L$, the total horizontal projection of the ...
-1
votes
2answers
2k views

Injective function: example of injective function that is not surjective.

Does there exist an injective function that is not surjective? Could I have an example, please?
2
votes
2answers
78 views

Inverse of $f(x)=x^3$ function

please help me :can we inverse this function: $f(x)=x^3$? I know that if a function is a bijective function only then it can be inversed. Is this a bijective function?
0
votes
1answer
54 views

Extensions of continuous bounded functions

If $u:U \rightarrow \mathbb{R}^{n}$ is bounded and continuous can $u$ always be extended such that $u \in C(\bar{U})$? and is $u$ uniformly continuous?
1
vote
3answers
74 views

Why aren't multi valued functions invertible?

I recently learnt that functions are invertible if and only if they are bijective. But why aren't multi-valued surjective 'functions' invertible?
1
vote
0answers
753 views

help with function word problem

A heated piece of metal cools according to the function c(x) = (.5)x - 7, where x is measured in hours. A device is added that aids in cooling according to the function h(x) = -x - 2. What will be the ...
0
votes
1answer
70 views

What denotes the essence of a function object in mathematics?

In other words, when does something become a function, and why? Take this, for example: x = y (x + z) = 350 Is anything enclosed within the brackets considered ...
3
votes
1answer
59 views

Does the function $f: \mathbb R \to \mathbb R^2, t \mapsto (t^3, t^2)$ have a history?

I heard one mathematician briefly mentioning that the function $f: \mathbb R \to \mathbb R^2, t \mapsto (t^3, t^2)$ is very famous and has a history. Do you know what was meant by that?
0
votes
1answer
12 views

Correct syntax for defining the vertex of the graph of a cuadratic function

I got into a discussion with my math teacher. Given the function: $$ f(x)=a(x-h)^2+k $$ If $h = 3$, $k = 2$ and $a = 1$, then: $$ f(x)=(x-3)^2+2 $$ We can agree that, when graphing the function, ...
3
votes
1answer
56 views

Painting $\mathbb R^+$ with two colors which sum of two same color numbers be the same.

Can any one paint $\mathbb R^+$ with two colors which sum of two numbers with the same color has the same color. Additional condition: Both colors should be used. I tried use Cauchy functions like ...
0
votes
1answer
35 views

Finding range of functions

I was wondering if there is a sure fire process of finding the ranges of functions without plotting them on a graph? At the moment (only for quadratics) I find the vertex of the graph through ...
1
vote
1answer
39 views

Mapping/function. Show that $M\subseteq f^{-1}(f(M))$.

Let $f:A\to B$ be a mapping. Show that for all subsets $M$ of $A$ satisfies \begin{equation} M\subseteq f^{-1}(f(M)). \end{equation} Give an example that $f^{-1}(f(M))$ doesn't need to be ...
0
votes
1answer
181 views

Expressing logarithms as ratios of natural logarithms

$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an ...
3
votes
1answer
68 views

Given $f(x) = x\log_{2}x$, how do I compute $f^{-1}(10)$?

Let $f(x) = x\log_2 x$. Compute $f^{-1}(10)$ to at least three decimal places of accuracy. Explain how you did this. Note: for a function $f:A\rightarrow B$ for which there is exactly one point $a$ ...
2
votes
1answer
78 views

Is this function one-to-one, onto, both or not a function?

$P(N) \times P(N) \to P(N)$ defined by $f(A,B)=A \cup B$. Answer: I gave a counterexample for one-to-one because if $A=\{\{\},\{1,2\},\{1\},\{2\}\}$ and $B=\{\{\},\{2\},\{3\},\{2,3\}\}$ then $A \cup ...
1
vote
2answers
61 views

Can every function which can be described by words, be formulated as well?

Almost one year ago i was amused when i saw this page. It was the generation of the prime numbers using the floor function, mostly. I became more interested about the things we can do with the floor ...
0
votes
2answers
37 views

Show that $(\phi_{n}^{(n)})^{-1}= -(\sum_{i=0}^{n-1}(\phi_{i}^{(n)})^{-1})$

So I have $n+1$ points $x_{0},x_{1},...,x_{n} \in \mathbb{R}$ and a following quasi-function: $\phi_{j}^{(n)}=\prod_{i=0,i \neq j}^{n}(x_{j}-x_{i})$ Show that $(\phi_{n}^{(n)})^{-1}= ...
6
votes
2answers
6k views

Proving a function is onto and one to one

I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding. To prove if a function is one-to-one, it says that I have to ...
1
vote
0answers
26 views

Linear programming: can someone explain how the time steps work here?

I'm reading a paper, "A Player Selection Heuristic for a Sports League Draft". In it, the authors have come up with a method to assist you in picking players for a fantasy sports league. I'm having ...
2
votes
2answers
110 views

Value of $ f(2012)$

$f(x) $ is an injective function . The definition of $f(x)$ is like following: $$ f:[0, \infty[\to \Bbb R-\{0\}, f\left(x + \frac{1}{f(y)}\right) = \frac{f(x)f(y)}{f(x) + f(y)} $$ If $f(0) = 1$ then ...
2
votes
2answers
346 views

Critical Values of a Function

I need to find the critical values of $h(t) = t^{3/4} - 2t^{1/4}.$ So I began by finding the derivative of the function and simplifying: \begin{align*} h'(t) &= (3/4)t^{-1/4} - (2/4)t^{-3/4} \\ ...
2
votes
2answers
550 views

Prove that f(x)=1/(1+x) is not uniformly continuous [closed]

How can I prove that $f(x) = \dfrac{1}{1+x}$ is not uniformly continuous on $(−1,\infty)$. Thank you.
0
votes
3answers
90 views

What are the methods to find approximatly the 5th roots of an equations

By which method, can I find the nearest root of : $x^5−2x+1.1=0$ ? Thank you.
13
votes
4answers
4k views

Can the inverse of a function be the same as the original function?

I was wondering if the inverse of a function can be the same function. For example when I try to invert $g(x) = 2 - x$ The inverse seems to be the same function. Am I doing something wrong here?
1
vote
1answer
61 views

Composition of function with it's inverse on subdomains

I have a short question. We have to check the following statements and tell for which one the equal sign holds. Let $M \subset \mbox{domain } f$ and $N \subset \mbox{Im } f$. ...
0
votes
1answer
75 views

An inverse question of uniformly convergence

{Edit: since I made some mistake on the pointwise limit and the uniformly continuous.} A classical results in elementary analysis state that if a sequence of continuous function $f_n(x)$ on $[0,1]$ ...