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

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5 views

Can notion of submodularity be extended to functions that are not proper?

I am familiar with notion of submodularity in functions whose range is $\mathbb{R}$. I was wondering if the notion extends to functions whose range is $\mathbb{R} \cup \{-\infty\}$.
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2answers
57 views

Prove or disprove: $\sup\left \{ x\in\mathbb{R}|x^{2}-5x+6\leq0 \right \}=3$

No homework:http://www2.mathematik.hu-berlin.de/~gaggle/S15/MATHINFO/UEBUNG/nachholklausur.pdf Prove or disprove: $\sup\left \{ x\in\mathbb{R}|x^{2}-5x+6\leq0 \right \}=3$ I would say the ...
3
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3answers
68 views

Taylor-polynomial of function $f(x) = e^{x}*\sin(2x)$

This is not homework, I'm asking to learn for an exam which I'll write in 2.5 months. Count the Taylor-polynomial 3th grade of the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x}*\sin(...
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3answers
53 views

Can a function have both a global maximum, and minimum? [on hold]

Can a function have a global maximum, and minimum?
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1answer
44 views

Solution to the convoluted integral equation

A have the following equation: $$f(a)=\int_0^ag(x)f(x)\,dx,$$ where $g(x)$ is a known function. Is there any solution to $f(a)$ just in terms of $g$?
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0answers
2 views

Terminology and existing work related to repeated function application (“annealing?”, canonicalization)

Suppose I have two functions, $f()$ and $g()$, where the image of one is a subset of the domain of the other and vice versa. There are some cases in which $f()$ and $g()$ are inverses of one another (...
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1answer
15 views

Proof of composite funtions question

Given $f:A \rightarrow B,\, g:B \rightarrow C,\,$ and $\,h: C \rightarrow D\,$ are functions. Show that $h(g(f(x))),\;h(g(f(x)))$ are defined and they are equal. Can $f(g(x))$ be defined? I know how ...
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2answers
30 views

Is this example being bijective?

There is a related nice problem discussing this (deeper discussion): Difference between bijection and isomorphism? But I do not want to ask it that further. This simple problem confuses me for a ...
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3answers
33 views

Find all local extremums of $f(x)=x^{2}e^{-x}$ and decide if these are global extremums

As all my other questions, this one isn't homework (it's preparation for an exam). I'd like to know if I did everything correctly. In my previous task, I had a mistake in the first derivation. But ...
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1answer
19 views

Behaviour of the solution of an equations as a function of a parameter

Let $x$ be the smallest positive solution of the following equation $$x=\frac{1}{\beta}\ln x+a-\frac{1}{\beta}\ln a+c$$ where $\beta>1$ and $a,c\in(0,1)$ are fixed constants. I need to know the ...
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3answers
4k views

When functions commute under composition

Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse. But then ...
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1answer
32 views

Find all local extremums of $f(x)=\frac{x}{x^{2}+x+1}$ and decide if these are global extremums

Did I do everything correctly? Find all local extremums of the following function and decide if these are global extremums (i.e. maxima or rather minima of the function on its entire domain) ...
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2answers
34 views

Are polynomial fractions and their reductions really equal? [duplicate]

I'm reading Larson's AP Calculus textbook and in the section on limits (1.3) it suggests finding functions that "agree at all but one point" in order to evaluate limits analytically. For example, ...
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1answer
39 views

Rearrangement of equations

Trying to find solutions to the equation of $$(1/x)+1=2\sin⁡x+3$$ and I have to rearrange it into the form of $x=f(x)$ as I am using the fixed point method and iterations. Struggling to find new ...
7
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3answers
267 views

Partial fractions and using values not in domain

I'm studying partial fraction decomposition of rational expression. In this video the guy decompose this rational expression: $$ \frac{3x-8}{x^2-4x-5}$$ this becomes: $$\frac{3x-8}{(x-5)(x+1)} = \...
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1answer
17 views

Feature transformation

Does someone know a computationally efficient bijective function $f$ : $\mathbb{R}\rightarrow (y_{0},y_{1})$ ? Preferably, $(y_{0},y_{1})=(-1,1)$ and $(0,1)$.
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1answer
25 views

Proving A = {1, 1/2, 1/3, 1/4,…}. Having trouble in showing onto.

Prove the set A = {1, 1/2, 1/3, 1/4,...} is infinite. I did by showing A is equivalent to N...
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votes
4answers
100 views

Sum to infinity of trignometry inverse: $\sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right)$

If we have to find the value of the following (1) $$ \sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right) $$ I know that $$ \arctan \left(\frac{4}{r^2+3} \right)=\arctan \left(\frac{r+1}2 \right)-\...
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0answers
29 views

Different Alternate Representations of Functions

Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions? For example, I know of two such ways 1) Taylor Series Expansion 2)...
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0answers
41 views

Prove that a certain hypergeometric function assumes either the value $\frac{1}{2}$ or 1

Numerics appear to indicate that the function \begin{equation} f(\alpha)= \end{equation} \begin{equation} \frac{\sqrt{\pi } 3^{-3 \alpha -1} \Gamma \left(2 \alpha +\frac{3}{2}\right) \, _5F_4\left(...
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1answer
26 views

Question about injective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $U$ is a set $g:T\rightarrow U$ is a function such that $g \circ f$ is injective then also $f$ is injective ...
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2answers
46 views

How to formally prove whether this function is onto or not?$ K(x) = x^ 2$ where $x \ge 0$.

$K(x) = x^2.$ The domain and range of this function comprise of non-negative real numbers. If it were real numbers instead of "non-negative" real numbers, then it seems easy to prove it by ...
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2answers
29 views

Calculate $P'(x)$ for $x \in (-1,1)$

$P$ is a power series with $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ Calculate $P'(x)$ for $x \in (-1,1)$ When I read this task (not homework), I got some questions: 1) ...
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1answer
60 views

“Of the order of” notation

I have a function where all terms have the same coefficient $x^3$ in it: For example $f(x) = ax^3 - bx^3$ Can I say that in big $O$ notation: $f(x) = O(x^3)$ $f(x) = O(x^3)$ as $x \rightarrow 0$ ...
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1answer
30 views

help with finding roots using excel or matlab

I've been asked to finding the roots of the equation $x^3-6x+2=0$. They've called it the 'pivot method' but I can't seem to find the 'real' name of it. I've looked at secant, newton, bisection etc... ...
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1answer
13 views

Data/Feature normalization

Let's say I have a set of random elements of the interval $(-1,1)$. $$S=\{0.03,-0.1,0.5,-0.45,...\}$$ I'm looking for a bijective function $f(x)$ which normalizes the elements of $S$ such that the ...
2
votes
2answers
52 views

Proof that $P(x)=x-\frac{1}3 x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$

Proof that $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$ First of all, I need to convert this to a series: $$\sum_{k=1}^\infty \frac{x^{2k-1}(-1)^k}{...
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4answers
52 views

For what $k$ is $f(x) = kx^2-2x+k$ negative for all values of $x$?

What are the values of $k$ for which the quadratic function $f(x) = kx^2-2x+k$ is negative for all values of $x$? The values of $k$ should definitely be negative.
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3answers
63 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a,\,b >0$ and $a<b,\,$ then we have to find $\lim_{x\to \infty} f(x)$ I tried as follows]1 But at end I got stuck .
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1answer
15 views

How to find the correct operation to fit a limited set of operations?

I'm trying to find a math operation indicated by -> which will result in y given x. x -> y 0 -> 0 1 -> 6 2 -> 5 3 -> 4 4 -> 3 5 -> 2 6 -> 1 ...
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0answers
20 views

When can you “plug in” a function $g(x)$ directly into a taylor expansion of a function $f(x)$ to get the expansion of $f(g(x))$, specifics below

I have asked a couple of questions Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to ...
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1answer
26 views

Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
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0answers
34 views

Does Taylor series around point zero (maclaurin series) always exist for differentiable function?

For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...
0
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1answer
30 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
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2answers
44 views

Function is continous

If functionis continous at x=0 the we have to find the value of k I got a solution , but I am not able to understand what they have done in second step . Can anyone explain me
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2answers
40 views

Is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative?

Assume a function $f(x)$ is differentiable, is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative? $$\max\left(\frac{...
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votes
2answers
258 views

What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
0
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1answer
44 views

How accurate is the approximation of the number of rough numbers?

A number is called a $y$-rough number, if it has no prime divisor below $y$. The number of rough numbers in an interval, lets say, $[10^{99},10^{100}]$ is approximately the length of the interval ...
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1answer
24 views

Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
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0answers
41 views

On the vanishing of integrals involving the $\sinh$ function. [on hold]

Suppose for some positive real $\theta$ that $$\int_1^\infty f(x)\sinh(\theta\log \sqrt x) \mathrm{d}x = 0$$ Where $f(x)$ is a non-constant and continuous function of $x$. What necessary properties ...
2
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1answer
32 views

How is it called if two functions have the same order?

Lets have $f(x_1)>f(x_2)\implies g(x_1)>g(x_2) \forall x_i \in \mathbb{R}$. Is this property between $f$ and $g$ named in some way?
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2answers
77 views

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$.

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$. There is no example like this in my math book.
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3answers
304 views

If $f(g(x)) = g(x)$, what is $f(x)$?

If $f(g(x)) = g(x)$ then can we conclude $f(x)=x$? (In fact, $g(x) = x + 2f(x)$.) Or which properties should $g(x)$ satisfy? (like one-to-one, etc.)
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2answers
44 views

Function that is onto and continous

Can there be a continuous onto function $f : \mathbb R \to \mathbb R \setminus \mathbb Q$? I know if a function existed it would map reals to the irrationals. Any ideas?
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2answers
55 views

Taylor-polynomial of $f(x)=\log(\cos(x))$

$f: (-\frac{\pi}{2}, \frac{\pi}{2}) \rightarrow \mathbb{R}, f(x) = \log(\cos x)$ Count the Taylor-polynomial $T_{2}(f, 0)(x)$ of the second degree of $f$ in $x_{0} = 0$ Alright because it was ...
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6answers
223 views

Prove that $\lfloor\frac{n+1}{2}\rfloor+\lfloor\frac{n+2}{4}\rfloor+\lfloor\frac{n+4}{8}\rfloor+\lfloor\frac{n+8}{16}\rfloor+ \dots=n$

Prove $$\left[\dfrac{n+1}{2}\right]+\left[\dfrac{n+2}{4}\right]+\left[\dfrac{n+4}{8}\right]+\left[\dfrac{n+8}{16}\right] + \dots=n$$ where $[x]=\lfloor x\rfloor$ $$$$ It was suggested that ...
9
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5answers
843 views

APICS Mathematics Contest 1999: Prove $\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$ is a constant function of $x$

This is question 3 from the APICS Mathematics Competition paper of 1999: Prove that $$\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$$ is a constant function of $x$...
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1answer
18 views

Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
7
votes
3answers
4k views

Functions similar to Log but with results between 0 and 1

I need a function similar to Log but it should produce numbers between 0 and 1 Something like: f(0)=0 f(1)=0.1 f(2)=0.15 f(3)=0.17 f(100)=0.8 f(1000)=0.95 f(1000000000)=0.99999999 I need this in ...
3
votes
2answers
59 views

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = - f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck