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

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2
votes
0answers
44 views

If $f(2x-f(x))=x$ . Find all bijective functions.

It is given that $f :[0,1] \rightarrow [0,1] $ and it is bijective. If $f(2x-f(x))=x$ , find all such f. Is my solution correct? My attempt $f(x)$ is bijective. thus there exists g(x) which is the ...
1
vote
1answer
41 views

What happen to composite of infinite number of continuous functions?

We all know that a composite of continuous functions is continuous. And this holds for any $\textbf{finite}$ number of functions. My question is what happen to infinite number of functions? Is it ...
-2
votes
0answers
26 views

What should it be called please suggest me.

Given two functions $T1: S\rightarrow U$ and $T2: U\rightarrow V$, who do we read the composition $T2\circ T1$? By "read", I mean in the sense that "$A\subset B$" is read "$A$ subset $B$" or "$x\in ...
0
votes
2answers
43 views

When $\ln(1+y) = y + o(y)$?

I was reading a proof which utilize the fact that: $\ln(1+y) = y + o(y)$ http://math.stackexchange.com/a/842557/160028 I'm not so sure what is the meaning of $\ln(1+y) = y + o(y)$. When is it ...
5
votes
4answers
294 views

Can functions be defined by relations?

So let us say that for whatever reasons, we are not allowed to use function symbols in first-order logic. Then can we define and use a function only by relations?
0
votes
1answer
32 views

Finding the “canonical decomposition” of a function — I don't know if I'm doing it right

I've been told to identify the terms in the canonical decomposition of the function r |-> exp(2*pi*i*r) from R -> C. I've been able to give an answer, but I think i might have misinterpreted the ...
1
vote
1answer
46 views

Differentiation of the Beta function

I suppose that \begin{align*} \frac{\partial}{\partial x}\left[B\left(x,y\right)\right]=&\frac{\partial}{\partial x}\left[\int_0^1t^{x-1}(1-t)^{y-1}dt\right]\\ ...
0
votes
0answers
16 views

show that it is an increasing function by derivative [duplicate]

how can we prove that "power or generalized mean" is an increasing function by derivative? and so it can show the relationship between means ( comparison of means) like arithmetic mean and geometric ...
0
votes
1answer
29 views

Multivariable-calculus, derivative and second derivative [closed]

I got the function $f(x,y)=\ln \sqrt{x^2+y^2}$. The task is to find the derivative function and the second derivative function. How do I get there?
0
votes
1answer
20 views

Factor the equation either by pairs method or any other [closed]

I tried to split by pairs but i got nowhere. $$x^2 - 4y^2 - 4x + 4$$
1
vote
2answers
23 views

Showing a function is uniformly continuous by using the derivative

Let $f(x) = \sqrt x \cos\frac{1}{x}$. Show $f$ is uniformly continuous on $(0, \infty)$. Now as I recall, we've learned in class that if $f'(x)$ is bounded therefore $f$ is uniformly continuous. ...
3
votes
2answers
40 views

Example of a function $F(x,y)$

I'm trying to find a non trivial function $F(x,y)$ such that $div F(x,y)=0$ everywhere and $F(x,y)=0$ on the unit square. I know that there are some books that provide such example but I didn't find ...
2
votes
3answers
63 views

show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$

Let $f:I\rightarrow \Bbb{R}$, differetiable three times on the open interval $I$ which contains $[-1,1]$. Also: $f(0) = f(-1) = f'(0) = 0$ and $f(1)=1$. Show that there's a point $c \in (-1, 1)$ ...
0
votes
2answers
43 views

Multivariable-calculus

The task is the attached image. We got the function and the domain of definition. The task is to decide the function´s lowest value and biggest value plus the range. Lowest value should be -sqrt3,6 ...
-1
votes
2answers
35 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
71 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
1
vote
3answers
41 views

The inflection points of $f(x)=(x^2-4x+1)e^{-x}$

I got the function $f(x)=(x^2-4x+1)e^{-x}$. The task is to find the inflection points. The correct answer is $x=4-\sqrt{5}$ and $ x=4+\sqrt{5} $. I got the second derivative to $f(x)$. But when I ...
1
vote
2answers
39 views

Multivariable-calculus. Find the stationary point (critical point)

I got the function $f(x,y)=\ln(1+x^2+y^2)$. The task is to find the stationary points. The correct answer is $(0,0)$. How do I find the stationary point? I`ve differentiated it at put it equal to zero ...
0
votes
2answers
42 views

Combinaision of two functions

Let us denote $X_0 = \{x, y\}$ and $X_1 = \{a, b\}$ two disjoint sets of variables; let us denote $V$ a set of values. I have two functions $f_0 : X_0 \rightarrow V$ and $f_1 : X_1 \rightarrow V$, ...
6
votes
4answers
692 views

generalized way of finding minimum value of a function?

$f(x)=\frac{x^{2}-1}{x^{2}+1}$ for every real number for $x$, the minimum value of $f$ is what? How can I find the minimum value of this function.I only know trial and error method, but it's not a ...
2
votes
1answer
36 views

Bijection between natural numbers $\mathbb{N}$ and natural plane $\mathbb{N} \times \mathbb{N}$

I know that is possible to build a bijection between the set of natural numbers $\mathbb{N}$ and the natural plane (the cartesian product of $\mathbb{N}$ by itself, $\mathbb{N} \times \mathbb{N} = ...
0
votes
0answers
20 views

Bounded function in 2 variables

hi guys i have 3 limits and i "solved" them by showing that all of them are 0 by the rule 0 * Bounded function = 0 buy i said "solving" because i cant show those function are actually bonded but i ...
1
vote
2answers
120 views

Show that $f(x)=x^4$ is convex

for $x\in (0,\infty)$ show $f(x)=x^4$ is convex. I know it is convex since $f''(x)>0$ . How can we show by using definition? do we have to use Let L be linear space. $t\in[0,1],y\in ...
-1
votes
1answer
49 views

Do the graphes of $f(x^2)=\sin(x^2)$ and $g(x)=\sin(x^2)$ look the same?

Do the graphes of $f(x^2)=\sin(x^2)$ and $g(x)=\sin(x^2)$ look the same? If this is true then: $f(x^2)=\sin(x^2)=g(x)$ and thus the $x^2$ in the parenthesis of $f$ does not matter at all???
1
vote
1answer
69 views

How to graph $y=f(x^2)=\sin(x^2)$?

How to graph $y=f(x^2)=\sin(x^2)$? I have substituted as follows: $$\begin{cases} y=f(a)=\sin a\\ a=x^2\end{cases}.$$ Then if I graph this with the coordinate axes $y$ and $a$ I get the ordinary ...
2
votes
1answer
44 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
vote
1answer
40 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 ...
3
votes
4answers
549 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
votes
0answers
22 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
votes
0answers
6 views

How can I find a hyperbolic function denoting zoom levels?

I'm working between two values. The first ($m$) represents the number of meters wide an estimate of location accuracy is, and the other ($z$) represents a vague level of zoom as described below. This ...
0
votes
1answer
67 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
votes
0answers
53 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 ...
0
votes
0answers
23 views

Function where circle is special case with i=2

I saw this function some while back but cannot recall its name and therefore cannot find it and research it. Can someone please remind me its name? $|x|^i + |y|^i = 1$ where $i$ is ${1,2,3,4,5 ...}$ ...
6
votes
2answers
110 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
votes
3answers
39 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
votes
1answer
56 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 ...
2
votes
3answers
74 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
votes
1answer
43 views

proving that specific function isn't continuous

$d(x) = Dirichlet Function $ $f(0) = 1 $ $f(0)$ is continuous at $x = 0$ $g(x) = d(x)f(x)$ . I need to prove(in two ways: with delta and epsilon, and with Arithmetic of limits) that if $f(x)$ is ...
0
votes
1answer
18 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
votes
2answers
98 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 ...
-4
votes
2answers
50 views

Combining two equations to make a function? [closed]

So, I know that $x^2 - (\frac{x}{x-1})^2 = 5$ and I know that my function is $f(x) = \sqrt{9-x^2}$ but I'm not sure how to include the x from the first equation into the function to write it in one ...
2
votes
1answer
58 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
votes
0answers
14 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) - ...
1
vote
4answers
110 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
5answers
129 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: ...
-2
votes
1answer
19 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 ...
7
votes
1answer
200 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
53 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 ...