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

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6 views

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$.

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. Forgive me, this is probably really simple. Thanks.
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0answers
9 views

Is there a way to reverse engineer an already large number to make it smaller?

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...
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0answers
10 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
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1answer
30 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
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0answers
56 views

how to solve this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
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1answer
20 views

Is tangent monotonically increasing?

According to wolfram a function is monotonic if its derivative never changes sign, but the derivative doesn't have to be continuous. So I feel the answer is Yes, tangent is monotonically increasing. ...
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2answers
28 views

Prove that there is C such that $f(c)=0$

$$f\in C[a,b] $$If given that for every $ x\in[a,b]\text{ there exists } y\in[a,b]$ such that: $$|f(y)| \le \frac{1}{2}|f(x)| $$ Prove that there exists $ c\in[a,b]$ such that : $$f(c)=0$$ What ...
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0answers
15 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
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5answers
81 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
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1answer
23 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
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1answer
10 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
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0answers
14 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
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1answer
23 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
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2answers
26 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
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1answer
16 views

Composite of functions with absolute value range

I have a really big problem with this next task, determing the range of composite g(f(x)). $f(x)=-2x+4$ $g(x)=\lbrace{x^2-4};|x| \le 2\rbrace$ $g(x)=\lbrace{4x^2-x^4};|x|>2\rbrace$ The result ...
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2answers
40 views

Minimum and maximum of a two variable function

I have to study the type of critical points of the function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$ and find minimum and maximum on the generic circle centered in $ (0,0) $ and radius $ r>1 $. I ...
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0answers
32 views

Help with Proofs (Even and Odd Functions) [on hold]

How do you prove: (My attempts below) 1.Sum of 2 even functions is even ----------F(x)=f(-x) +g(-x) ---------- = f(x) + g(x) 2.Difference of 2 odd functions is even ----------F(x)=f(-x)-g(-x) ...
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0answers
23 views

Writing $\mathrm{SO}(2)$ as the zero-set of a function

Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is: $\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ ...
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0answers
6 views

Finding the revenue for a certain amount of units with a demand function.

First, I have no idea if that title makes sense at all... The problem I'm trying to figure out is asking; The price p and the quantity x sold of a certain product obey the demand equation: p=-1/9x ...
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5answers
86 views

limit of $f(x) = \lim \limits_{x \to 0} (\frac{\sin x}{x})^{1/x}$ [on hold]

Any ideas how to calculate this limit without using taylor? $$f(x) = \lim \limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{1/x}$$
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1answer
38 views

Piece wise function continuity [on hold]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
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1answer
46 views

Find all functions $F(x)$ for which $F (x) + F ((x − 1)/x) = 1 + x$

Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$. This ...
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2answers
49 views

Proof on Functions /Set Theory

Let $S$ be the set of all numbers of the form $a + b\sqrt 2$ where $a$ and $b$ are rational. Let $f : S \to R$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x$ and $y$ in $S$. Then $f(x)=f(1)x$ ...
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1answer
38 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
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2answers
32 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
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1answer
32 views

Is onto function necessarily a function?

The standard definition suggests that every element in the codomain should have a preimage. So, Can different elements in codomain or range have same domain? A worst question I think. Please reply... ...
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2answers
26 views

Banach contraction theorem exercise

Use Banach contraction theorem to find a solution of x+e^x=0 correct to 3 decimal places. Any suggestions how I should begin? Banach contraction theorem is new to me.
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2answers
49 views

How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
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3answers
46 views

Derivative of given $f(x)$ at $x=0$

If given this function: $$f(x) = \begin{cases} e^x, & x \le 0 \\[2ex] -e^{-x}+2, & \text{x > 0} \end{cases} $$ How do I calculate the derivative at $x=0$? Shall I calculate by the normal ...
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0answers
27 views

Does there exist such a function $f(x)$ that $f(f(…(f(x))))=\left (1-\frac {1}{\sqrt[n]{x}} \right)^n?$

Let $n=11...1$ (1996 figures). Does there exist such a function $f(x)$ that for all real $x \not =0, x \not =1$ holds $$f \left ( f\left (...\left (f(x) \right) \right) \right)=\left (1-\frac ...
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0answers
17 views

Transforming parts of functions

I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$ If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, ...
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1answer
37 views

Function with infinite maxima and minima [on hold]

Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval? Edit: This question came to me as I was reading on the dirichlet ...
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0answers
20 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
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1answer
18 views

Function Equivalent to the Maximum Operator?

All numbers are real, WLOG positive. $A + B + ... + N = T$ and $A' + B' + ... + N' = T$ I'm trying to figure out some function, f, such that if $f(A,B,... ,N) > f(A',B',...,N')$ then, ...
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1answer
60 views

What is the inverse function of $y=x^2 + 3x +2$? [on hold]

What is the inverse function of $f(x)=x^2 + 3x +2$? Please show your solution method and demonstrate that $f(f^{-1}(x))=x$
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0answers
17 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
2
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1answer
32 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...
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1answer
17 views

Prove that $f'(1)\ge n \left ( 1+ \sqrt[n]{f(0)} \right)^{n-1}$

Let $f(x)=(x+a_1)(x+a_2)...(x+a_n)$, where $a_1,a_2, ..., a_n -$ non-negative numbers. Prove that $$f'(1)\ge n \left ( 1+ \sqrt[n]{f(0)} \right)^{n-1}$$
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2answers
30 views

Domain and range of $f(x)=\arcsin[e^{-x}]+ \arcsin [e^x]$

What is the domain and range of $f(x)=\arcsin[e^{-x}]+ \arcsin [e^x]$ where $[x]$ denotes greatest integer function?
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2answers
75 views

$\lim_{n \to \infty} \int_{0}^{n}(1-\frac{3x}{n})^ne^{\frac{x}{2}}dx$=?

$$\lim_{n \to \infty} \int_{0}^{n}\left(1-\frac{3x}{n}\right)^ne^{\frac{x}{2}}dx$$ I thought about using the theorem of monotonic convergence and had ...
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0answers
6 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
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1answer
7 views

Colon and equals under product operator

I'm trying to understand the following equation: $\prod_{i:y_i=1} p(x_i) \prod_{i:y_i=0} (1 - p(x_i))$ The part I don't get is the subscript below the product operator. Does the $i:y_i=1$ under the ...
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1answer
15 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
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1answer
48 views

Construct a non-monotone continuous function of bounded variation

Construct a continuous function of bounded variation on $[0,1]$ which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function (somewhat). For example, at the ...
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2answers
53 views

Can every one variable equation be solved without graphing? [on hold]

Can every one variable equation be solved without graphing? How would you solve the following without graphing: $$3y + 4\sqrt{1-y^2} = 2$$
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0answers
17 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
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0answers
10 views

Why does $y(s)$ continuous imply that $f(s)$ with $f_l (s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum \max\{0,z_l(s)\}}$ is continuous?

Let $z:\triangle^{L-1}\to \mathbb{R}^L$ be continuous. Define $f:\triangle^{L-1} \to \triangle^{L-1}$ be defined component wise as $$ f_l(s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum_{l=1}^L ...
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1answer
27 views

Complex Numbers in Factoring [on hold]

Why does "$i$" only get involved in factoring a function when there is a ($+$) in the equation? EX: $x^2 + 9$.
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2answers
50 views

If $f(x-4) = x^3 + 2x^2 + 7x + 1$, determine $f(x)$.

I was faced with a pretty tricky math problem from my school. Here is the question. If $f(x-4) = x^3 + 2x^2 + 7x + 1$, determine $f(x)$. My first instinct is to graph the initial equation and shift ...
5
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0answers
37 views

Why was the zeta function introduced?

I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function'). I also know the 'Zeta Function' $\zeta(s)$ is defined as: ...