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

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2
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2answers
40 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...
1
vote
0answers
19 views

Is a function with a random variable continuous?

I often like to fool around on graphing calculators when I am bored. A function that can be very amusing is $f(x) = rand \times sin x$ Now, on my TI-84 Plus, this looks obviously discontinuous ...
18
votes
14answers
2k views

Produce unique number given two integers

Given two integers, $a$ and $b$, I need an operation to produce a third number $c$. This number does not have to be an integer. The restrictions are as follows: $c$ must be unique for the inputs ...
0
votes
0answers
9 views

A nonlinear version of Jamiołkowski-Choi isomorphism

I am not very sure whether this question is correct at all. We know that the completely positively maps between matrix algebras can be written as a Kraus sum, i.e. if $\phi:M_d \rightarrow M_d$ is ...
1
vote
0answers
13 views

Multidimensional fitting of two data sets

My problem is the following: A laser gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives ...
1
vote
2answers
100 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
0
votes
1answer
82 views

The number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$ (Putnam 1993)

Let ${\cal P}_n$ be the set of subsets of $\{1, 2, \dots, n\}$. Let $c(n, m)$ be the number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$. Prove that ...
0
votes
1answer
36 views

$f(x) = x^2 + 1$ maps $\mathbb R$ onto $[1, \infty)$

Let $f(x) = x^2 + 1$ and $x$ in $\mathbb R$. Prove that $f$ maps $\mathbb R$ onto $[1, \infty)$. Let $\mathbb R$ be the domain of $f$ and $[1, \infty$) the codomain of $f$. Then $f$ maps ...
0
votes
1answer
249 views

Cancellation laws for function composition

Okay I was asked to make a conjecture about cancellation laws for function composition. I figured it would go something like "For all sets $A$ and functions $g: A \rightarrow B$ and $h: A \rightarrow ...
0
votes
1answer
35 views

A function with positive Hessian at a critical point, without having a minimum there

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
4
votes
1answer
49 views

Existence of a function with boundary conditions for derivatives

Does there exist a function $f\in C^2(\Bbb{R},\Bbb{R})$ such that $\frac{f'(x)}{f(x)}\rightarrow_{x\rightarrow\infty}+\infty$ but $\frac{f''(x)}{f'(x)}\rightarrow_{x\rightarrow\infty} 0$ ? I know for ...
4
votes
5answers
129 views

How to work out miles between Longitude values based on a Latitude value.

We know that when Latitude is 0, the distance between Longitude values is roughly 69 miles. When the Latitude is +/-90, Longitude values are 0 miles. At 0 Latitude, the earths circumference is ...
0
votes
2answers
22 views

Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
2
votes
2answers
32 views

The domain of $f(x)=\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$

$$f(x)=\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$$ It's obvious that this: $$\frac{1-\cos(x)}{1+\cos(x)}\geq0$$ and this $$1+\cos (x)\ne0$$ $$x\ne\pi+2k\pi, k\in Z$$ are the conditions of the function's ...
0
votes
2answers
35 views

Proving $f(x) = x^2 + 1$ is surjective

Let $f(x) = x^2 + 1$, where $x$ is a real number. Prove that $f$ maps $ \mathbb R$ onto $[1, \infty)$. We must show that if $y \in Y$, then there exists an $x$ such that $f(x) = y$. I am tempted ...
2
votes
5answers
134 views

Is it true that $f: S\to S$ be a function: $(f \circ f)$ is bijejective if, and only if $f$ is bijective?

Let $f :S\rightarrow S$ be a function. Show that $f\circ f$ is bijective if, and only if, $f$ is bijective. My solution. If $f\circ f$ is bijective if, and only if, $f$ is ...
1
vote
1answer
33 views

Elevation of 3D function

$f(x,y) = \begin{cases} x^2/y & y \neq 0 \\ 0 & y = 0\end{cases}$ I need to draw the elevation (or you may call it Equivalent curve) of this function and I don't know how to draw them. Can ...
0
votes
2answers
2k views

Mirror a function about x = c axis

I'm trying to mirror a function $f(x)$ about the $x=c$ axis. To mirror it about the $x=0$ axis you just have to plot $f(-x)$. I tried to mirror $f(x) = x^2$ about the $x = c$ axis. And I found that ...
0
votes
0answers
32 views

Is a plane minus a line a region?

According to ODE by Tenenbaum and Pollard, a region is defined as follows: Each point of the set is the center of a circle whose entire interior consists of points of the set. Every two points of ...
0
votes
1answer
36 views

Functions operating in uncountable sets with cardinality $\gt\aleph_1$

A generic function $y=f(x)$ maps a number in the set of real number $X$ in another number in the set $Y$. It's well known that the irrational numbers are not countable. It's also known we can get a ...
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votes
2answers
2k views

Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...
2
votes
1answer
117 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
vote
2answers
915 views

Understanding the IMDb weighted rating function for usage on my own website

I'm trying to implement a review function on my website, but I want it weighed. I checked on IMDb and they have this: weighted rating $(WR) = (v / (v+m)) R + (m / (v+m)) C$ where: $R$ = average for ...
0
votes
1answer
34 views

Definition of Kronecker Delta

Is $\delta _{mn}=1$ when $m\neq n$, and $\delta _{mm}=0$? I am not very good at Math. So would you give me the answer and explanation please?
2
votes
1answer
33 views

Finding the range of $f(x) = 5-x$

Let $a < b$ and $f: (a, b] \to R$, $f(x) = 5 - x$. What is the range of the function? How do you find the range of an equation with unknowns?
1
vote
1answer
43 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
0
votes
0answers
14 views

What happens when scaling a rectangle using a pivot point?

With multitouch screens, you can pinch to zoom. When such a gesture is triggered you are supplied with: An x scale factor; A y scale factor; A x pivot point; A y pivot point. When I have a ...
1
vote
1answer
26 views

Function on plane with incenter

Let $f$ be a function from the set of all points on the plane to the nonzero real numbers. Suppose that for any triangle $ABC$ with incenter $I$, we have that $f(I)=f(A)f(B)f(C)$. What are the ...
1
vote
1answer
36 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
0
votes
1answer
286 views

What can I deduce from the graph of marginal value?

Homework problem: We have a graph of value: x coordinate number of widgets made. y coordinate value in $. For the first 5 widgets the value greatly increases to $100 and then after 5 ...
1
vote
1answer
24 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
3
votes
4answers
122 views

What is the the $n$ times iteration of $f(x)=\frac{x}{\sqrt{1+x^2}}$? [closed]

We were asked to determine the composition $f \circ f \circ f \circ...\circ f $, $n$ times, where $f(x)=\dfrac{x}{\sqrt{1+x^2}}$. Anyone has an idea?
2
votes
1answer
35 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
0
votes
1answer
25 views

How to prove subsets of a function

$ f: \mathbb R \to \mathbb R$ is defined as a one to one function. For any collection of subsets $A_1, A_2, A_3 ......A_n$ prove that, $$ f(A_1 \cap A_2 \cap A_3 ......A_n) = f(A_1) \cap f(A_2) \cap ...
1
vote
1answer
39 views

Proving f(rx) = rf(x)

What's the difference between the proofs of $$ f(rx)= rf(x) \forall r \in \mathbb Z ,\forall x \in \mathbb R $$ and $$ f(rx)= rf(x) \forall r \in \mathbb Q , \forall x \in \mathbb R $$ where $ f : ...
0
votes
2answers
44 views

Matrix of Functions

I was thinking about a problem and I realized for a family of functions say $L^1([a,b])$ one can define matrices with functions as elements: $$ A=\left[ \begin{array}{ccc} f_{11}& ...
0
votes
1answer
20 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
0
votes
1answer
20 views

Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
2
votes
2answers
55 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
0
votes
1answer
20 views

Even function about a point over a restricted range

Why is $f(x)=(x-1)^2$sin$(n\pi x)$ even about $x=1$ for $0\leq x \leq2$? I understand that $(x-1)^2$ is even about $x=1$ and I can plot the graph for various values of $n$ on wolfram alpha, but how ...
1
vote
2answers
14 views

Setup Quadratic Word Problem

I need help setting up this quadratic word problem, I have no idea where to start. Among all pairs of (real) numbers whose sum is 17, find a pair whose product is as large as possible. What is the ...
0
votes
1answer
42 views

Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
0
votes
2answers
58 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
1
vote
2answers
106 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 ...
3
votes
1answer
278 views

Exact value of expression

Let $$f(x)=\frac{4^x}{4^x+2}$$ and $$S=\sum_{n=1}^{2005}f\left(\frac{n}{2005}\right)$$ What is the exact value of $S$? I tried to write $a=4^{\large\frac{1}{2005}}$, then ...
1
vote
2answers
54 views

Graph a function

I have a question, I have a function: $$f(x) = \frac{-x^2-10x}{2}$$ I'm really confused how to replace the x. So, what would be the points in $y$ if $x$ were: $-4, -3, -2, -1, 0, 1, 2, 3, 4$?
1
vote
3answers
30 views

How many solutions to quadratic logarithms?

For a given Log equation with a quadratic $x$, such as $$F(x)=\log(x^2)$$there appear to be two $x$ values, for every $F(x)$, a positve and a negative. However, if $F(x)$ is ...
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
5
votes
2answers
36k views

Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?

I would very much like to have a complete list of the types of polynomial functions. I know that theres: ...
2
votes
1answer
67 views

If a function is both upper and lower semicontinuous, does it have to be continuous?

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] ...