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

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Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
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1answer
12 views

showing that if a function is a bijection, then there exists a an identity function

Let f:x-y be a bijection, show that foi =iof =f where i is identity function. I know that a bijection is one which is bith noe to one and onto. The problems is that the question is so trivial that I ...
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1answer
15 views

Figure out simple joint formula for sets of vectors?

I have on the left side four pairs of sample values that result in the respective pair of values on the right side. ...
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2answers
24 views

How do you shift a sigmoidal curve to the right?

How do you shift the function $1$ $/ ( 1 + e ^ {-x} )$ to the right without altering the shape of the curve?
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0answers
12 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
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1answer
11 views

Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
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0answers
14 views

Quick way to determine the number of horizontal asymptotes

I understand how to calculate horizontal and vertical asymptotes, both by using the trick of comparing the degrees of the numerator/denominator and by using calculus. What I would like to know is ...
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2answers
22 views

Inverse Function: unique?

Is it true in general that the inverse of a function is unique if it exists? Why is this so? Clearly inverses in groups are unique. However, that seems not directly applicable in this case...
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1answer
53 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
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0answers
12 views

On sequence and function [on hold]

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non negative integers. Let $A_k$ (respectively $B_k$) be the sets of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of inteers such ...
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0answers
6 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
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9 views

what is the use of tilde in specifying domain of functions

what does the tilde "~" in (-3,-2)~{-2.5} mean with respect to domain? when we generally say that the function has its domain from A to B, and excludes C, we can write it as (A,B)-{C}. my ...
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2answers
21 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
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2answers
21 views

Rules for combination of odd vs even functional equations

Let $f$ be an even function, and $g$ odd. Let $h$ be some arbitrary function. Is it the case that $f(x) + h(x),\ fh(x),\ hf(x),\text{ and }f(x)h(x)$ are each even or odd according to $h$, and that ...
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2answers
17 views

Finding the Asymptote / Root of a reciprocal function

$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as the ...
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0answers
48 views

Why is the inverse of the Devil's Staircase not measurable?

I recently did an exercise to show that a monotone function $f:X→ℝ $ is Borel measurable (it even only asked for Lebesgue measurability). On the other hand, the inverse of the Devil's Staircase ...
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0answers
18 views

How to integrate this function and decide lambda

I want to decide lambda for I also want to integrate the same function. Please help!
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0answers
18 views

Is the function space of a Banach space with product topology a Frechet–Urysohn space?

Consider $E^X$ the space of functions $f: X\to E$ where $X$ is a set and $E$ is a Banach space over $\mathbb{R}$ or $\mathbb{C}$. Using the product topology, then if a sequence of function $f_n \in ...
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0answers
25 views

differentiate the given function. Simplify your answers [on hold]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
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0answers
10 views

What is the formula for single frequency generation function obtained from FFT?

What is the correct formula of a function that generates specific tone from fourier transform? I thought that having: transformata - an array with FFT of a source sample. v = transformata[freq] - ...
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1answer
19 views

Functions (Finding Inverse)

$f(x) = x^2 + 2x$ , domain ${x ≥ 1}$ Question: find the inverse The inverse is $f(x) = 1 + \sqrt(1+x)$ (taking the positive square root only) As $f^{-1}(5) = 2$ and as 2 is an element from the ...
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1answer
12 views

Functions inverse + domain

Question part a): !([http://imgur.com/sKJbFKu]) Answer: !http://imgur.com/jRfeXkW Can anyone explain why the inverse must be the negative square root?
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1answer
35 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
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2answers
121 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
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3answers
41 views

Why is the inverse of this function not a function?

Why does $F^{-1}$ need to be defined on all of $Y$? I can have this function: $g(x)=x,\quad x\ne 3$ and even though it is not defined for all $x$ in its domain, it is still a function, right?
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1answer
18 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
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1answer
23 views

Something basic; why do I get two different bounds on $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$?

Let $n$ be a natural number. Let $f(x) = \frac{x^2}{\sqrt{x^2 + n^{-1}}} + \sqrt{x^2 + n^{-1}}$. since $x^2 + n^{-1} \geq x^2$, it follows that $$|f(x)| \leq \frac{x^2}{|x|} + \sqrt{x^2 + ...
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2answers
25 views

Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
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0answers
21 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [duplicate]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
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2answers
37 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
1
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1answer
16 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
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3answers
31 views

invertibility of $f^{-1}$

In my introductory maths book there is a statement (it follows a theorem) that says:"Note that if f is one-to-one, then $(f^{-1})^{-1} = f$, and so $f^{-1}$ is invertible and also one-to-one because ...
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1answer
27 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
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1answer
39 views

Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$… show $f$ preserves order.

Spivak Calculus, 4th ed., problem 3-17: If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
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3answers
38 views

Functions definition + question

Am I correct in saying that for Functions, the below is the correct definition: For each value of x in the domain there is only one value of y in the range. Hence, the picture below means that it is ...
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1answer
33 views

Showing a function is decreasing

I have $$a_{n} = \left|\int^{(n+1)\pi}_{n\pi} x^{-p}\sin{(x)}~\mathrm{d}x\right|$$ and want to show this is monotonically decreasing, how would I do this? Note $n\in\mathbb{N}$ and $p>0$.
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1answer
39 views

How can a function not be one to one and be a function?

My understanding of the definition of a function Given any x, there is only one y that can be paired with x My understanding of a 1 to 1 function Given any y, there is only one x that can be paired ...
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2answers
37 views

Range for the function $f(x) = 3x + 2$ with domain $x > 0$

The function below is defined for continuous domains Sketch the graph and state the range of the function Question: $f(x) = 3x + 2$ for the domain $\{x \in \mathbb{R} : x > 0 \}$ The straight ...
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2answers
56 views

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$

Problem : Solve $|x^2+x-4| =|x^2-4| +|x|$ We can find the critical point of each modulus function individually then we get : $x =\pm 2;$ and $x = 0$ $x = \frac{-1 \pm \sqrt{17}}{2}$ So there are ...
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1answer
23 views

Roots of Taylor's series.

Show that there is exactly one value of x which satisfies the equation $$2cos^2 (x^3+x)=2^x+2^{-x} $$ I solved this using Taylor's series: $$2^x+2^{-x}=2\{1+\frac {x^2 \{ln2\}^2}{2!}+\frac {x^4 ...
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1answer
30 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
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1answer
21 views

For any function $f$, $f(s) \in f(S) \not\implies \Leftarrow s \in S$

This already contains many counterexamples, so I'm not seeking any more of them; I'm interested in learning about my errors with the notation and definitions. Richard Hammack P213 Defintion 12.9: ...
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0answers
27 views

HELP! Marginal Probability Density Function of X and Y (pictures).

I understand the formulas for finding the marginal PDF of X and Y, however, in this example, how do we get from this to that: Thanks a lot!
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2answers
51 views

Please advise on the order of calculation

I have to make a software making the calculations below over some set of data. That is basically not a problem. The problem I have is with notation of the second formula which is the (Utility ...
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0answers
38 views

Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [on hold]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection $h:A\rightarrow ...
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1answer
45 views

Equating coefficients

Excuse me,i don't know how to deal with this problem,i try it for all time of last night, this equation is on "Concrete Mathematics" page 200: d(n) is the number of derangements. e^z is the ...
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0answers
10 views

Discrete Math Trace recursive function

Does anybody know how to trace this function by specifying the recursive calls to the function? The inputs are: A = {24, 15, 7, 10, 8, 30}, i = 2, n = 6 RandomElement(A) returns an element of A ...
2
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1answer
39 views

What is the use of iterating over a function?

If we have a function, say: $$ f(x) = 3x $$ We can get output values based on linearly increasing input: $$ f(1) = 3(1) = 3 $$ $$ f(2) = 3(2) = 6 $$ $$ f(3) = 3(3) = 9 $$ $$ ... $$ Or, we can ...
2
votes
1answer
62 views

$\textbf{Q}$ fails to prove some correct $\forall$-rudimentary sentence

Show that the existence of a semirecursive set that is not recursive implies that any consistent, axiomatizable extension of Q fails to prove some correct $\forall$-rudimentary sentence. I ...
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1answer
18 views

Convert sum to function

I need to convert $\sum_{i=0}^N \frac{C_1}{C_2+C_3i}$, to a function $C_1$, $C_2$ and $C_3$ are constants. I am interested in resulting function itself and method as well.