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

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1answer
47 views

Find $f(x^2)$ of $f(x)$

How can I find $f(x^2)$ of $f(x)$? For example: I take the function $f(x)=a$ where a is an algebraic expression like $\sin x$, $3x^3$, etc. Now, is it possible to find $f(x^2)$ of $f(x)=a$? If it ...
1
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1answer
102 views

Is this function Lipschitz in two dimensions?

I want to show that the function $A(x,y)$ is Lipschitz in two dimensions. The function is defined as follows ...
2
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4answers
1k views

How do I show that f is strictly decreasing on (0, infinity)?

I have been asked to define $f: (0, \infty) \to (0, \infty)$ by $f(x) = \frac 1 x$ a) How do I show that f is strictly decreasing on $(0, \infty)$? I realize that I have to show that $f'(x)<0$, ...
0
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0answers
28 views

Is there accepted notation and/or terminology for the smallest cover of $S$ with cells from $P$?

Let $X$ denote a set. Then for $S \subseteq X$ and $P$ a partitioning of $X$, define $P \diamond S$ as the smallest cover of $S$ with cells from $P$. Explicitly: $$P \diamond S = \bigcup\{Q \in P ...
0
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1answer
81 views

Separable Function: Alternative Representation

How does one get the following function $$ f(u) = f(x+iy) = \frac{u^{z-1}}{e^{-u}-1}, $$ where $z$ is a constant complex number and u is a complex variable, into the form: $$ f(x+iy) = v(x,y) + ...
3
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2answers
83 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
8
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4answers
277 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
2
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3answers
41 views

$f(a)=b$ where $a$ and $b$ are algebraic expressions of $x$

Is it possible to write functions in the form $$f(a)=b$$ where $a$ and $b$ are algebraic expressions of $x$ (e.g $a=3x^2$ , $b=4x^5$)? The example function would be: $$f(3x^2)=4x^5$$ Do these ...
2
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1answer
88 views

Find the all function if $2f(mn)\ge f(m^2+n^2)-(f(m))^2-(f(n))^2\ge 2f(m)f(n)$

QUestion: Find all the function $f:N\to N$, such for any $m,n\in N$, have $$2f(mn)\ge f(m^2+n^2)-(f(m))^2-(f(n))^2\ge 2f(m)f(n)$$ This problem is from Mathematical olympiad 2014(chongqing ...
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3answers
83 views

Finding the range of values of $\frac {\sqrt{1+x^2}-1}{x}$

How do I find the range of the above expression, given that $x \ \in \mathbb{R} -\{0\}$ A seemingly useful method is substituting $x=\tan \alpha$ If $y$ be the given expression, then following my ...
0
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1answer
77 views

Proving that a specific function isn't continuous

Assume the following: $d(x)$ is the Dirichlet Function, $f(0) = 1$, $f(0)$ is continuous at $x = 0$, and $g(x) = d(x)f(x)$. I need to prove (in two ways: with $\delta$ and $\epsilon$, and with ...
0
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1answer
32 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
3
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2answers
101 views

If $f( \cos^2(x) ) = \cos^2(x)$ can I assume that $f(x) = x$?

I am new to functions and domains and I am not sure that I can assume following because I think that the range of first function is $[0, 1]$ and the range of second is $(-\infty, \infty)$. The ...
2
votes
1answer
82 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
0
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0answers
29 views

Explicitly relating two functions containing exponential terms

I have two functions related to the distribution of administered drugs in the body: $$\begin{align}c_1(t) &= a_1\exp(-k_{11}t) - b_1\exp(-k_{21}t)\\ c_2(t) &= a_2\exp(-k_{12}t) - ...
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4answers
130 views

$f(x^2)$ even or odd

I've been working on the following example: Is the following even, odd or neither: $f_{0}(x^2)$, where $f_{0}(x)$ can be any unknown function I've tried the following: 1) for example I assume ...
6
votes
4answers
151 views

If $f\left(x-\frac{2}{x}\right) = \sqrt{x-1}$, then what is the value of $f'(1)$

Find $f'(1)$ if $$f\left(x-\frac{2}{x}\right) = \sqrt{x-1}$$ My attempt at the question: Let $(x-\dfrac{2}{x})$ be $g(x)$ Then $$f(g(x)) = \sqrt{x-1} $$ Differentiating with respect to x: ...
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1answer
43 views

Question about writing cyclometric function in function of x

I have an excercise about cyclometric functions and I'm stuck right now: $\cot(2*arcsec(x))$ Let $\mathbb y=arc\ sec(x) \Leftrightarrow sec(y)=x$ then $cot(2y)=\frac {cos(2y)}{sin(2y)}=\frac ...
9
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1answer
226 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?
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1answer
116 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
61 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
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0answers
38 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
108 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
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0answers
40 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 $\lim_{x\to +\infty}=a<1$.
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2answers
53 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
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1answer
85 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
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1answer
44 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
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1answer
35 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
245 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
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2answers
142 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
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1answer
49 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
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2answers
152 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
47 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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2answers
158 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: ...
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16answers
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
70 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 ...
2
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2answers
90 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 ...
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1answer
51 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
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1answer
38 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
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1answer
66 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 $f(D)$?
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1answer
33 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
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1answer
106 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
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2answers
56 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
72 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
29 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
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2answers
79 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
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1answer
51 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
298 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
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3answers
39 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
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0answers
21 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 ...