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

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7
votes
1answer
201 views

finding the value of $f(2001) $ if…

if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
1
vote
1answer
91 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
2
votes
1answer
55 views

Given that $f(x) = x + \frac{1}{x}$ where $x>1$, find $f^{-1}(x)$

Given that $f(x) = x + \frac{1}{x}$ where $x>1$, find $f^{-1}(x)$. I don't understand and how to start. Please help.
0
votes
0answers
30 views

Simplify $L_{-1}(x) + I_1(x) $

Is there a simple solution for x << 0 of the following equation: $$Y(x) = L_{-1}(x) + I_1(x) $$ Where $L_{-1}(x)$ is modified Struve function and $I_1(x)$ is modified bessel function. For ...
6
votes
1answer
98 views

Some conditions to obtain that $\int_1^{x}e^{f(t)}dt\sim_{x \rightarrow +\infty}\frac{\exp(f(x))}{f'(x)}$

Playing with the function $e^{t^2}$ I conjectured the following result : Let $f\in C^2(\Bbb{R},\Bbb{R})$, assume that : $f'(x)\rightarrow_{x \rightarrow +\infty}+\infty$ ...
0
votes
0answers
35 views

Examples of functions with this property

Can you give some example of functions with this property? $f:\mathbb{R}^+\to \mathbb{R}^+ $, $f^{\prime}>0$, $f^{\prime \prime}<0$, $f(0)=0$ and $\lim_{x\to +\infty}=a<1$.
1
vote
2answers
44 views

Implicit functions - understanding [closed]

$(y)^{0.5} + y = x^3 + x$ Is $y$ implicit function of $x$, for any $x$? For which $x$ values the $y$ function will be implicit?
8
votes
1answer
58 views

Is every real valued function on an interval a sum of two functions with Intermediate Value Property?

If $I$ is an interval of real numbers , then is it true that any function $f:I \to \mathbb R$ can be written as $f=f_1+f_2$ , where $f_1 , f_2 : I \to \mathbb R$ have the Intermediate value property? ...
1
vote
1answer
36 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 ...
1
vote
1answer
23 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 ...
2
votes
2answers
52 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 ...
0
votes
2answers
30 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?
1
vote
1answer
40 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
1answer
87 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
0answers
33 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
44 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 ...
0
votes
2answers
122 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: ...
23
votes
15answers
3k 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 ...
4
votes
1answer
55 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 ...
1
vote
1answer
50 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
23 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
40 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 ...
1
vote
1answer
28 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
26 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 $f(D)$?
0
votes
1answer
27 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
42 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
45 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}& ...
2
votes
2answers
56 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
22 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
15 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
44 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: ...
3
votes
1answer
288 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
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 ...
2
votes
1answer
135 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
1answer
13 views

Function that has potential to increase current value, based on current value

Math is not my strong suit, let's start there. (Be gentle.) I have a game engine that is "ticking" every 1 second. I would like for a number, A, to increase at an ...
4
votes
2answers
97 views

Why is the period of $f$, $\pi$?

I came across a problem, which asked to compute the period of the function $$f(x)=3^{\sec^2x-\tan^2 x}.$$ The answer provided was $\pi$. I don't get how.
0
votes
4answers
35 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
1
vote
1answer
31 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
2
votes
1answer
69 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] ...
0
votes
1answer
34 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
0
votes
1answer
27 views

Finding a population Function

I have been given the population of the USA from 1790 - 1980 (increasing in intervals of 10) and I am asked to solve this differential equation. Using t as time in yearrs P as size of population at ...
0
votes
2answers
63 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 ...
0
votes
0answers
44 views

Function's graph sketch

$f(x) = x+9$ $g(x) = 2x-3$ I need to draw the sketch of $\min(f,g)$ and $\max(f,g)$. I tried: http://sketchtoy.com/62375984 The yellow is the min and the blue is max. The lines should be ...
0
votes
2answers
42 views

Getting a diverse set of three numbers from two numbers

I'm using this information to build an interface to pick a color, but I feel that this question is purely math-related. Please correct me if this is the wrong StackExchange site for this. I am making ...
0
votes
2answers
57 views

Geometric meaning of results obtained in (a) and (b)

The task: Plot the function $\sqrt{1-x^2}$. What does it look like? What is the geometric meaning of the results you obtained in (a) and (b)? Can anybody help me with geometric mean? I can't ...
1
vote
2answers
39 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
0
votes
1answer
30 views

How to make clear a letter is a function?

How should I make clear that a symbol is a function? Usually a function is denoted by the letter $f$ or $g$, or is directly applied to arguments (e.g. $c(x,y)$) or is implied to be a function by an ...
0
votes
2answers
62 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
vote
2answers
45 views

Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this. I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one ...