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

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0
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1answer
22 views

Decompose to even and odd functions

Suppose we have the function $f(x) = |x-1|$. I have to find the even and odd parts of the function and write them in terms of Heaviside Function. I have no idea what should I do here? I tried and it ...
-5
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0answers
16 views

Prove that this function is surjective

Hey guys can you help me ? I need to prove that this function is surjective ( the one with m and n ). Sorry I can't type in Latex. Here's a link for a picture http://i.imgur.com/7saD78v.jpg
0
votes
1answer
20 views

How to prove local minima are global?

I have the function $f(x,y) = (x^2 - 4)^2 + y^2,$ which has two local minima at $(2,0)$ and $(-2,0).$ How can I prove that these are global minima?
4
votes
2answers
247 views

I can do the math but not the problems, help? [on hold]

So whenever my instructor / teacher is going over the notes and teaching the new lesson to the class, I listen to what he says, take notes, and do the practice problems along with him. Often times I ...
0
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2answers
36 views

What is the graph of this?

I need know what is the graph of y=y(x), where: $\sqrt {{{(x - \sqrt 5 )}^2} + {y^2}} + \sqrt {{{(x + \sqrt 5 )}^2} + {y^2}} = 6$ , thank you
0
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0answers
46 views

Does there exists a bijection between $\mathbb{R} $ and $\mathbb{R} \backslash A$?

Let $A$ denote the set of algebraic numbers. Prove that $ \lvert \mathbb{R} \rvert = \lvert \mathbb{R} \backslash A \rvert$ I know A is countable, and I think that it shall be easier to show the ...
0
votes
1answer
11 views

solve for the max of the sum of two points on a function a given distance apart?

I just thought of this concept and am not very experienced in math, so I'm assuming there's an easy solution I'm overlooking. For a given function y = f(x), how can one find the maximum value for the ...
0
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4answers
58 views

What function is on the graph?

I need to know what function is on the graph. And how do I determine the function name by its graph?
2
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4answers
269 views

Is the function continuous

$$f(x)=\begin{cases} x^{2}-1 & x \leq 1\\x- \frac{1}{x} & x \geq 1\end{cases} $$ How can you show if these kind of functions are continuous or not?
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2answers
43 views

How do you solve these recurrence relations for a closed form?

I'm not sure what methods are used to solve recurrence relations for a big-$O$ notation. Thinking about the problem conceptually doesn't really help me, but I feel like I could use some form of ...
1
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2answers
24 views

Pre-images proving question

Let $f : X → Y$ be a map. For any subset $C ⊆ Y$ (of the codomain of $f$ ), the preimage of $C$ under $f$ or the $f$ -preimage of $C$ is defined as the subset (of the domain of $f$ ) $f^{−1}(C) := ...
1
vote
1answer
19 views

At what point does normal line intersect curve second time?

At what point does the normal line to $y=-5+4x+3x^2$ at $(1,2)$ intersect the parabola a second time? $y'=6x+4$ $m_{tangent}=6(1)+4=10$ $m_{normal}=-\dfrac{1}{10}$ $y=f'(1)(x-1)+f(1)$ ...
0
votes
4answers
31 views

Why is my reasoning wrong in determining how many functions there are from set A to set B?

I am trying to count how many functions there are from a set $A$ to a set $B$. The answer to this (and many textbook explanations) are readily available and accessible; I am not looking for the ...
0
votes
1answer
21 views

Inequality involving $f(x)=x^p$ with $0<p<1$

Let $f(x)=x^p$ with $0<p<1,p\in \Bbb{R}$ defined in $[0,+\infty[$. Knowing that $f$ is crescent, show that, $\forall a,b\in \Bbb{R}$, $$(|a|+|b|)^p\leq|a|^p+|b|^p$$
3
votes
2answers
41 views

Limits to infinity, even and odd functions

I have a couple of questions regarding a practice test I just made, so the subject might vary a little bit but most of it has to do with limits. $$ \lim_{x \to \infty} \dfrac{7x+3x^2}{1-x^3} $$ ...
0
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0answers
12 views

Is this operation valid when doing time-complexity analysis?

I am analyzing the time-complexity of an algorithm that uses binary-search to solve the 3SUM problem. I am asked to give the big-oh and big-theta timings. \begin{align*} \sum_{i=1}^n \sum_{j=1}^i ...
0
votes
1answer
15 views

Shifting the domain

Wee have a function $f$ with a domain $[0,2]$ and range $[0,1]$. What is the domain and range of $f(4-x)$? The range is obviously the same, but I don't get why the domain is $[2,4]$. $f(4-x) = ...
0
votes
1answer
23 views

Prove that f has at least one global minimizer

$f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function such that $\displaystyle\lim_{\|x\| \to \infty} f(x) = \infty$ On a side note: how can a function have more than one global minimizer? Is a ...
2
votes
1answer
95 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$ such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent while $\,f^{(n)}=f$. Could you give me some hints? I truly ...
0
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0answers
25 views

Where is $f(x)=\sqrt{|1-x^2|}$ Lipschitz continous?

It seems to me that the Lipschitz constant is 1 near $x=\pm 1$, $y= \pm 1$ $$ |f(x)-f(y)| \leq \frac{|x+y|}{\sqrt{|1-x^2|}+\sqrt{|1-y^2|}}|x-y| $$ How would you define the Lipschitz constant L?
1
vote
1answer
34 views

What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
0
votes
3answers
55 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
5
votes
1answer
68 views

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
0
votes
2answers
23 views

help finding the inverse of this function

$$f(x)=x^3 - \frac4x$$ Find the value of the inverse for $x=6$. The answer is $2$ and I'm having problems finding the inverse function because there are two variables of different powers.
0
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0answers
16 views

Composite function of a multiple condition function.

Q. $f\left(x\right)=1+x\:for\:0\le x\le 2$ $f\left(x\right)=3-x\:for\:2<x\le 3$ Determine $g\left(x\right)=fof=f\left(f\left(x\right)\right)$. Am a bit confused with the domain and range of ...
0
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2answers
38 views

Max-min inequality

It is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?
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0answers
15 views

Complex Plane - Analytic Function

I am trying to understand the definition of an analytic function and how to solve for it's domain. I understand that for $f(z) = {1\over z}$ the function is analytic on the complex plane except for 0. ...
-1
votes
1answer
29 views

Need help with an algorithmic function [on hold]

Consider the following claim: for any positive constant c, f(cn) ∈ Θ(f(n))? Either show the claim is true or give a counterexample.
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2answers
37 views

Need help ordering a list of functions

List the functions below from lowest order to highest order. If any two or more are of the same order, indicate which. $n$, $n^3$, $2^n$, $\ln n$, $n^2$, $\ln^2 n$, $\sqrt n$, $2^{n−1}$, $\ln n$, ...
0
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0answers
21 views

Continuous functions using interval notation

Let $f(x) = \sqrt{x-2}$. Use interval notation to indicate where $f(x)$ is continuous. I don't know how to solve it.
0
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1answer
27 views

f is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^−1 (V ))$ = V prove?

$f$ is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^{−1} (V ))$ = $V$ This is an assertion and i said it was true. But i am confused as to what is referred to as the domain and range in this question. I would say ...
3
votes
1answer
58 views

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$ I don't know how to proceed, any help would be really appreciated..
0
votes
1answer
23 views

If $f: B \rightarrow C$ and $g: A \rightarrow B$ be two functions let $h = f \circ g$. Then for $h$ to be onto what can we say about $f$ and $g$?

Let $f: B \rightarrow C$ and $g: A \rightarrow B$ be two functions and let $h = f \circ g$. Given that $h$ is an onto function, which one of the following is TRUE? (a) $f$ and $g$ should both be ...
1
vote
1answer
36 views

What is the series of the function 3 / ( 1- x^4)

I know that $f(x) = \frac{1}{ 1-x } = \sum_{n=1}^\infty x^n$. We can find that $g(x) = \frac{1}{ 1-x^4 } = \sum_{n=1}^\infty (x^4)^n = \sum_{n=1}^\infty x^{4n}$. Does the sum converge? what is the ...
2
votes
1answer
61 views

Is $f(x) = \infty$ a function?

Recently, while solving a problem where a certain set of functions $f:\mathbb Z^+ \rightarrow \mathbb Z^+$ had to be found given a number of conditions, I noticed that $f(n)=\lim_{a\to+\infty} a$, ...
2
votes
1answer
33 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
1
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2answers
42 views

Functions and limits two directions

A really cunning question which I can't seem to solve. If $\lim _{x\to \infty }\left(f'\left(x\right)\right)=0$ does it necessarily means that $\lim _{x\to \infty ...
0
votes
0answers
56 views

Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
1
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2answers
30 views

Converting Recursive Function into Closed/Explicit Form

so I have this recursive function here: $\forall n>1,f(n) = 2(f(n-1)) + n-1$, (where it is $0$ when $n$ is less than $1$) So I have tried to use iteration for this but it just gets more ...
1
vote
1answer
16 views

generalization of midpoint-convex

Let f : (a,b) → R is a midpoint-convex function (I didn't say continuity). Here I'd like to verify following inequality ""directly"". f( (x1+x2+x3)/3 ) ≤ (f(x1)+f(x2)+f(x3))/3 .. I can easily ...
1
vote
1answer
40 views

How to prove this? “For all sets A,B⊆D and functions f:D→R, we have f(A∩B)⊆(f(A)∩f(B)).” [duplicate]

Here's my attempt: f(A∩B) = f({x|x∈A∧x∈B}) = {f(x)|x∈{x|x∈A∧x∈B}} f(A)∩f(B) = f({x|x∈A}) ∩ f({x|x∈B}) = {f(x)|x∈{x|x∈A}} ∩ {f(x)|x∈{x|x∈B}} = {x|x∈{f(x)|x∈{x|x∈A}}∧x∈{f(x)|x∈{x|x∈B}}} And now I'm ...
1
vote
2answers
31 views

Do we simplify the Proof by Contradiction?

Prove the following by contradiction: Suppose $a,b\in\mathbb{Z}$. If $4|\left(a^2+b^2\right)$, then $a$ and $b$ are not both odd (in other words, $a$ and $b$ are even) So, I did this: Assume $a$ ...
0
votes
3answers
35 views

Find real functions knowing their values for all natural point

Consider a function $f(x) : \mathbb{R} \rightarrow \mathbb{R}$ smooth enough such that $f(\mathbb{N}) \subseteq \mathbb{N}$. Is there some methodologies to find another function $g(x): \mathbb{R} ...
1
vote
4answers
44 views

Proving onto and 1-1 functions

I understand the 1-1 function side of things, but I still don't really get how to prove that the function is onto Question: Prove that the function $f:\mathbb{R}-\{2\} \to \mathbb{R}-\{5\}$ defined ...
-1
votes
3answers
61 views

Does f(x) = g(u)?

If $f(x) = x + \sqrt{2-x}$ and $g(u) = u + \sqrt{2-u}$ is it true that $f = g$? I squared both sides $\sqrt{x + \sqrt {2-x}} = \sqrt{u + \sqrt {2-u}}$ $\sqrt{x} + 2-x = \sqrt{u} + 2-u$ I then ...
0
votes
1answer
52 views

Graphs of a functions: $ e^{x^2} , e^{1/x} $

I don't understand how to plot similar functions without a calculator. 1. $\arctan {1\over x-2} $ 2. $e^{x^2} , e^{1/x}, e^{2x\over1-x^2} $
3
votes
2answers
50 views

How do I read this definition of injective in English?

This is a different but related question to one I asked earlier. I link to it here: "To show that f is injective" - I don't get this statement I am pretty new to "functions" having ...
0
votes
0answers
21 views

verifying function one to one and onto

f: z-> z x z such that f(n) = (2n, n+3). Verify one to one or onto? I tried y = n+3 = f(n) Condition for one to one f(x) = f(y) so , x+3 = y+3 therefore , x = y Another try , If I take n = 1 an ...
2
votes
0answers
30 views

subspace of the Vector Space of real valued functions

This is a problem from Hoffman and Kunze's Linear Algebra 2nd edition. I am trying to determine whether or not a particular subset of the set of all real valued functions is a subspace. I've done ...
1
vote
1answer
30 views

Function Equivalent to a Constant Paradox

Say I define $z(x,y) = x^2+y = \text{constant}$ Then $\left(\dfrac{\partial z}{\partial x}\right)_{y} = 2x$ However, $\left(\dfrac{\partial \text{ constant}}{\partial x}\right)_y = 0$ Shouldn't ...