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

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1answer
9 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
10 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 ...
0
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0answers
18 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
14 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
38 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
29 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
21 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
5 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
81 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 ...
2
votes
1answer
44 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 ...
2
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2answers
48 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 $f(x)=f(1)x$ ...
1
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1answer
35 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) ...
3
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2answers
28 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
30 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
25 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
26 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
36 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
19 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
17 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?
6
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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
51 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
16 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 ...
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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: ...
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2answers
78 views

Suppose that $f(0)=f(2\pi)$. Show that there exists an x such that $f(x)=f(x+\pi)$.

I am supposed to show that there exists an $x$ in the interval $[0,\pi]$ such that $f(x)=f(x+\pi)$ by considering another function $g:[0,\pi] \to \mathbb{R}$ defined by $g(x)=f(x)-f(x+\pi)$. Should I ...
1
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1answer
21 views

Is proving this with Taylor is true?

Prove for every $x \neq 0$ in the domain $(-1, \infty )$ that: $$ln(1+x) < x $$ My prove: $$ln(1+x) < x \iff $$-$$ e^{ln(1+x)} < e^x \iff $$-$$ 1+x < e^x $$ and since Taylor's ...
2
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2answers
44 views

Find all functions so that $f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})}$ [on hold]

I have to find all functions so that $$ f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})} $$ I have no idea how to solve this one. Any help would be appreciated!
2
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2answers
31 views

Prove a function approaches infnity when the deriviative is greater than $0$

Here's my question: Let $f$ be a function which has a derivative in $\Bbb R$ such that $f'(x)\geq0$ and $f''(x)\geq0$ for all $x \in \Bbb R$ Prove that if there is some $a \in \Bbb R$ such ...
1
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4answers
57 views

Show that a continuous function f either has a root hence $f(c)=0$ or $|f(x)|> e$ for $e>0$.

From what I understand, I am being asked to show that a function $f$ on an interval $[a,b]$ either has a root $c$ such that $f(c)=0$ or it does not have a root hence $|f(x)|\ge e$. Should I apply the ...
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1answer
39 views

Intersection points between two functions

How can i find the intersection points between $$y=\frac{\pi}{4} x,\:\: y=\tan^3(x)$$ Thank you for your help. I know that the first (the obvious one...) is $0$. But how can i get another one?
0
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2answers
39 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
2
votes
7answers
93 views

Evaluating $\lim_{x\to1}{\frac{\sqrt{x^2+3}-2}{\sqrt{x^2+8}-3}}$ without L'Hospital's Theorem

I've been trying to evaluate$$\lim_{x\to1}{\frac{\sqrt{x^2+3}-2}{\sqrt{x^2+8}-3}}$$ I tried: (a) Rationizing the numerator -> Error (b) Rationizing the denominator -> Error (c) Factoring out $x$ ...
-1
votes
1answer
29 views

Homeomorphism between topological spaces defined by $f(x) < g(x)$

So, I have two continuous functions $f(x)$ and $g(x)$. $f,g : \mathbb{R} \longrightarrow \mathbb{R}$ and $f(x) < g(x)$ for all $x$ real. I have to show that $\{(x,y)\in \mathbb{R} | f(x) \leq y ...
1
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1answer
42 views

How to calculate $\lim_{x \rightarrow 0} \frac{\int_0^{G(x)} \arctan(s+2s^2) ds}{x^2}$ based on the following assumption?

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. To start this question, we need to integrate $\arctan(s+2s^2)$ but how do you do that? Then, what do we do ...