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

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2
votes
1answer
42 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
0
votes
2answers
37 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
0
votes
2answers
64 views

$\lim_{x \to \infty} f(x)=1 $ $\implies$ $f(x) \sin x$ is uniformly continuous on $\mathbb R$?

Let $f:\mathbb R \to \mathbb R$ be a continuos function such that $\lim_{x \to \infty} f(x)=1 $ , then is it true that $f(x) \sin x$ is uniformly continuous on $\mathbb R$ ?
1
vote
1answer
46 views

Intersection of Images of a function

I'm trying to understand intuitively why the image ( under some function ) of the intersection of subsets of the domain of that function is only contained ( and not equal ) to the intersection of the ...
9
votes
3answers
477 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
2
votes
1answer
18 views

How is it called when you apply min / max seperatly to each dimension?

I want to do the following: $$\begin{pmatrix}3\\1\\4\\1\end{pmatrix} = \min( \begin{pmatrix}4\\4\\4\\4\end{pmatrix}, \begin{pmatrix}3\\1\\4\\10000\end{pmatrix}, ...
2
votes
1answer
80 views

How to find all complex polynomial $f$ such that $1+f(z^n+1)=(f(z))^n$

Question: Let $n\gt1$ be a natural number. Is there a non-constant complex polynomial $P$ such that $P(x^n+1)=P(x)^n-1$ for all $x$? I saw this problem about polynomial, here is the question: Find ...
3
votes
2answers
64 views

Intuitive and convincing argument that functions are vectors

Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the ...
4
votes
2answers
35 views

What is a transformation?

I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld ...
0
votes
0answers
28 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
2
votes
1answer
56 views

Method for computing limit of a sin function as x tends to zero

I have a question about computing $$ \lim_{x \to 0} \sin\left(\frac{\pi x}{4|x|}\right)$$ I found the limit of $\pi x$ and $4|x|$ seperately and ended with $\sin(\pi/4)$ which is equal to ...
0
votes
3answers
58 views

Method for computing limit of a function as $x$ tends to zero

I have a question about computing $$\lim_ {x \to 0} \dfrac{(2/x^3)+(1/x^2)+(1/x)+1}{(1/x^3)+1}.$$ I used a shortcut method of dividing by the highest power but I don't think that I can use this method ...
0
votes
0answers
52 views

Elementary Pigeonhole Principle Question

Is my reasoning here correct? If not, advice would be appreciated. Thank you for your time! We assume that $A$ is finite and $f: A \rightarrow A$. We show that $f$ is one-to-one iff $ran \ f = A$. ...
2
votes
0answers
37 views

notation for minimum and maximum?

I'm trying to figure out the correct notation for this situation for use in Machine Learning. I have various ratings (for texts): ...
1
vote
2answers
85 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
0
votes
1answer
24 views

Objective function with two variables

A factory produces jointly two articles, and it has the problem to decide their prices in order to maximize the monthly income, knowing that the demand d1 (in hundreds of units) of the first article ...
7
votes
5answers
496 views

Functional Notation.

I have some doubts regarding function notation: First If I present a function I write:$f(x)$ If I write it's inverse:$f^{-1}(x)$ So why doesn't$f(f(x))=f^2(x)$ Second If $\frac{df(x)}{dx}=f'(x)$ ...
0
votes
1answer
51 views

Limit of a function is unique [closed]

I have read the proof of this property. The uniqueness of a limit of a function: Spivak's proof I was thinking we can also prove this informally by using the definition of function. For a ...
2
votes
2answers
64 views

Proving that a function satisfying $|f(x)-f(y)| \leq |x-y|^3$ is constant [duplicate]

Let $\mathbb R$ be the set of real numbers and $f: \mathbb R \rightarrow \mathbb R$ be such that for all $x$ and $y$ in $\mathbb{R}$, $$|f(x)-f(y) |\leq |x-y|^3.$$ Prove that $f(x)$ is a constant. ...
5
votes
2answers
77 views

What is this notation for a function? I've never seen it written like this before.

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$ I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside? ...
1
vote
4answers
42 views

How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
1
vote
0answers
63 views

Is $f^n_b$ a surjection?

For the purposes of this question, let $\mathbb{N}_k = \mathbb{N}_{\geq k} = \left\{n \in \mathbb{Z} | n \geq k \right\}$ and $\mathbb{R}_+ = \mathbb{R}_{\geq 0} = \left\{x \in \mathbb{R} | x \geq 0 ...
2
votes
1answer
30 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
0
votes
2answers
46 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
4
votes
1answer
68 views

Is there a name for the property of a function f such that $f(x,y)=f(y,x)$?

As in the title: is there a name for the property of a function such that $f(x,y)=f(y,x)$. I don't know how to be clearer than that. I tried to look for symmetric property on Google, but without any ...
1
vote
1answer
60 views

Implementing a function in PARI/GP

I want to define a function: $$g(n)= \begin{cases} +1 & \text{if $n=1$},\\ +1 & \text{if $n$ is an odd indexed prime}, \\ -1 & \text{if $n$ is an even indexed prime},\\ (-1)^r & ...
2
votes
0answers
43 views

What is the curve's name for the “reciprocal” equation of a circle?

The equation of a unit circle is $$(x-a)^2+(y-b)^2=r^2$$ When the origin $$(a, b)=(0,0)$$ the equation becomes $$y=(1-x^2)^{1/2}$$ Naturally when this equation is plotted on graph paper we get a ...
0
votes
2answers
53 views

$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
0
votes
2answers
15 views

Simplify and Evaluate function of X

I am having problems understanding the answer to this question Let $f(x) = 1 - x + 4x^2 $ Evaluate $(f(x+h)-f(x))/h$ Any help would be much appreciated, Thanks
1
vote
1answer
19 views

Intervals on which function is increasing and decreasing

Let $p(x)=x^5-q^2x-q$ , where $q$ is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below. We compute ...
0
votes
1answer
22 views

Blend n number of values by distance

I have n number of values which each have a distance that determens how much of the amount that should be blended. I've tried to illustrate my problem visually: The blue numbers is the values, the ...
20
votes
6answers
2k views

Is there a name for the function $\max(x, 0)$?

Is there a name for the function $ \max(x, 0) $? For comparison, the function $ \max(x, -x) $ is known as the absolute value or modulus of x, and has its own notation $ |x| $
2
votes
1answer
40 views

$\tan(x), \cot(x)$ function properties

Does $\tan(x)$ and $\cot(x)$ has symmetry axis? (like e.g $\cos(x)$ at $\pi k$ for $k \in \mathbb{Z}$), I tried think in the direction that $\sin(x)/\cos(x) = \tan(x)$ and both of them have symmetry ...
1
vote
1answer
38 views

fundamental period of sum of two periodic functions

Is there some formula to find fundamental period of sum of two periodic functions both of whose fundamental period is known. If yes what is the proof and the formula
1
vote
3answers
77 views

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's ...
1
vote
2answers
31 views

Sketching graphs abs value functions

how do I go forward with sketching the graphs of the following two functions? i) $y(t)=|2+t^3|$ ii) $f(x)=4x+|4x-1|$ thanks in advance!
0
votes
2answers
52 views

What does let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$ mean?

Let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$. I'm supposed to prove that this statement is true or false, $$∀f ∈ F, \;∃g ∈ F\tag i$$ so that $g(f(1)) = 2$ But I'm not ...
0
votes
2answers
51 views

Explanation of $\dfrac {x}{0}=\text{undefined}$ in form of function [closed]

As we all know zero is bit different then all the other real number. I have seen $\dfrac {x}{0}=\text{undefined}$ I know that this is because answer will be infinite, which is not mathematical term ...
1
vote
3answers
60 views

Find the minimum value of $t^3+t^2-2t-2$ given that $t$ is greater than or equal to 2

The original question was to find the range of the function f defined by: $$f(x)=\frac {(1+x+x^2)(1+x^4)}{x^3}$$ for $x>0$ Evidently, differentiating is not very helpful. So I wrote $f(x)$ as: ...
0
votes
2answers
76 views

A function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}(x)=\frac{1}{f(x)}$

Does there exist a non trivial function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}(x)=\frac{1}{f(x)}$ ?
0
votes
1answer
36 views

Find maximum value of function

I have to analyse the following function $F(b, k) = \frac{ab(k-1)}{k(1-ab)}$, given $a$ is a constant, $0 < a < 1, 0 < b < 1$; $k > 1$ and $kb < 1$. So I need to find $b$ and $k$, ...
0
votes
2answers
22 views

Confusion with the vertical line test of functions

Recently i saw the following article on wikipedia to see whether a graph is a graph of a function or not. http://en.wikipedia.org/wiki/Vertical_line_test It states that To use the vertical line ...
2
votes
2answers
55 views

If a one-to-one function's inverse is the same what must be true of the graph of f?

As a followup to this question. I'm trying to determine what must be true of the graph of $f$ in these cases. I've examined the two functions $f(x)= x$ and $f(x)= \frac{1}{x}$ and I'm not seeing any ...
1
vote
0answers
12 views

Constructing a periodic piecewise (piecemeal) function in Maple.

I'm trying to make a piecewise function that will have period $12$. That is, it repeats every $12$ units across the $x$-axis. I managed to do one cycle successfully with ...
0
votes
0answers
67 views

What does $X:Y\to x(t)$ mean?

Relatively new into math and working my way into it. I need some help understanding the statement below. X:Y -> x(t). Can someone please help me with what does it mean? So just to clarify I have a ...
0
votes
4answers
50 views

Find domain and range of $(f \circ g)$ for $f(x)=\ln x$ and $g(x)=x^2−1$

Word for word: Consider the functions $f(x)=\ln x$ and $g(x)=x^2−1$, find the domain and range of $(f\circ g)(x)$ I think this is asking to find the domain and range of $\ln(x)^{2}-1$ and the ...
5
votes
1answer
89 views

A real analytic function that takes each value in $\mathbb{R}$ three times

I was inspired by this question: it is quite easy to prove that for any positive odd number $2m+1$ there exists a function $f\in C^{\infty}(\mathbb{R})$ such that ...
3
votes
4answers
187 views

Prove/disprove: if $\lim\limits_{ n\to\infty} f(n)=\infty$ then $\lim\limits_{ n\to\infty}f(f(n))=\infty$

Let $f(x)$ a continuous function on $\Bbb{R}$. Prove/disprove: If $\lim\limits_{n\to\infty} f(n)=\infty$, then $\lim\limits_{n\to\infty}f(f(n))=\infty,$ where the limits are taken over $n \in ...
1
vote
1answer
18 views

One set of functions larger than another set of functions?

This summer I've been slowly working through Halmos's Naive Set Theory. I'm not that far, but I know what lies ahead, which is proving that one infinite set is larger than another (the reals larger ...
0
votes
1answer
29 views

How to handle a function from a set of functions to another set of functions?

Given sets $X$ and $Y$ we denote the set of functions from $X$ to $Y$ by $\text{Fun}(X,Y)$. Let: $k,n \in \mathbb{Z}^+$ $X_1 = \{x_1,x_2\dots, x_{k+1}\}$ $Y = \{y_1, y_2, \dots , y_n\}$ Then, ...