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

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2
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0answers
53 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
0
votes
1answer
48 views

Interesting question regarding elementary functions

I had this question at a test for a job interview and since and I didn't solve it. Some time later i still can't figure it out, so any insight is helpful. You need to write a function $f(x)$ such ...
0
votes
1answer
76 views

f is measurable iff its coordinate functions are measurable

I am really struggling to connect the sets in $\mathcal{B}(\mathbb{R^n})$ and $\mathcal{B}(\mathbb{R})$. Both inclusions are causing me problems. This questions seems a lot harder than it looks.
0
votes
1answer
38 views

Find the number of vertices in the graph

Let $n\ge 1$ and $V_n = (\left\{ 1,2,...n \right\}\rightarrow\left\{ 0,1,2 \right\})$. Let us define $G_n = \left<V_n, E_n \right>$. $f,g$, are two vertices. They are connected iff: $$\left|\{ i ...
0
votes
4answers
42 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
1
vote
1answer
58 views

Handy way to find the $x$ value where $\sin x \cos \left( \frac{\pi}{2} \sin x \right)$ is maximum?

Like in the title, is there a handy way to compute the $x$ values for which the function $$f(x) = \sin x \cos \left( \frac{\pi}{2} \sin x \right)$$ reaches its maxima? The derivative is $$f'(x) = ...
0
votes
1answer
15 views

Sinus curve with elbows / round steps?

Can I calculate a sinus function that has kind of elbows / round steps in it ? Or if I could get hold on the second curve. I need one of these functions for some graphical design. How would the ...
1
vote
1answer
133 views

roots of sum of exponential functions

Could anyone point me in the right direction of finding the roots of equations of the form $$ \sum_{i=1}^n a_ie^{f_i(x)}, $$ where $a_i \in \mathbb{R}$ and the $f_i$ are each first degree polynomials ...
1
vote
1answer
338 views

If a function has a inverse that is well defined is it a bijection

If I have a function (binary relation), $f: X \to Y:x \mapsto y$ and I show that it is well defined and I show that its inverse is well defined. Then have I shown that $f$ is a bijection? (That it ...
6
votes
10answers
1k views

how to see the logarithm as the inverse function of the exponential?

I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ...
2
votes
1answer
62 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
0
votes
3answers
70 views

what is inverse of $y = 5 ^ {\ln x }$

Problem: What is inverse of $y = 5 ^ {\ln x }$ Solution: $$y = 5 ^ {\ln x }$$ $$ \log_5 y =\ln x $$ $$ e^{\log_5 y} = x $$ After that "I don't know , how to get answer $ x=y^ { \frac {1} ...
0
votes
1answer
2k views

Difference between horizontal and vertical line tests.

Trying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell you about the graph? If an equation fails the ...
18
votes
2answers
340 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
0
votes
2answers
29 views

How do we emphasize that $\displaystyle x\mapsto\frac{1}{f(x)-y}$ “makes sense” if we know $y\notin\text{im }f$?

Please take a look at the following function $$x\mapsto\frac{1}{f(x)-y}$$ where $f$ is "some other function". Suppose we know $y\notin\text{im }f$, i.e. the expression in the denominator "makes ...
1
vote
2answers
60 views

limit of $ f(n) = 100 \left (1 - \frac{1}{n}\right) ^{ n}$

So I was daydreaming about math (like I do frequently) and I came up with this question/riddle. Say you have a die. If you roll a 1 you lose, otherwise, you win. This die has n sides on it, and you ...
5
votes
1answer
196 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
10
votes
4answers
189 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
2
votes
1answer
47 views

Taylor series $\ln(2+x)$ centered at $x=2$

Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...
2
votes
2answers
60 views

Understanding η-conversion (Lambda Calculus)

Let $h \in A\rightarrow (B\rightarrow C)$ I'm trying to understand the following reduction: $$\lambda x\in A. \lambda y \in B.(h(x))(y) \\= \lambda x\in A.h(x) \\= h$$ Apprantly, this is done by ...
0
votes
3answers
47 views

$f:\mathbb X \to \mathbb X$ , $f^m(x) = f^n(x)$, f is onto, prove that f is one-one.

$f:\mathbb X \to \mathbb X$ $f^m(x) = f^n(x) $, m > n , m and n are integers. $f^m$ is just $f \circ f \circ \dots f$ m times. It is given that f(x) is onto. Prove that f(x) is one-one.
-1
votes
1answer
33 views

Show the image of this function

Show that the image of the function (no derivative) $$f(x)=\frac{x^2+x+1}{x}$$ is: $$(-\infty,-1]\cup[3,\infty)$$ I tried to prove that it was increasing range and decreasing in others (maximum ...
4
votes
1answer
86 views

A constant function

$f:\mathbb{Z}\to \mathbb{R}$ is bounded above and satisfies $$f(n)\le \frac{f(n+1)+f(n-1)}{2}$$ Does it follow $f$ is constant ? There was a dreadful typo in the previous question (in the previous ...
1
vote
2answers
231 views

A proof of constancy

$f:\mathbb{N}\to \mathbb{R}$ is bounded above and satisfies $$f(n)\le \frac{f(n+1)+f(n-1)}{2}$$ Does it follow $f$ is constant ? I assumed $f$ achieves a maximum $M$ , suppose $n_0$ is the smallest ...
0
votes
1answer
17 views

When will function output have a specific decimal component

Given a function f(x), is there any way to predict when the function will give a specific decimal part without brute-force iteration over possible x-values? Ex. f(x)=100/a^2 with arbitrary a, when ...
0
votes
1answer
35 views

Sum of a function involving $n-$root

I'm trying to find the series of $$f(x)=\sqrt{1-x^{3}}$$ Can I just use the fact that $$\frac{1}{1-x}=\sum x^{n},\quad|x|<1$$ writting $x^{3}$ in the place of $x$ and then, getting this: ...
3
votes
3answers
234 views

finding the explicit function of a recursive sequence

So I have the recursive sequence $f(0) = 0, f(n+1) = 2f(n)+ (n+1)^2$, and I'm not quite sure how to make it explicit. Substituting $n$ for $n+1$ cleans it up a little, yielding $f(n) = 2f(n-1) + n^2$, ...
0
votes
0answers
38 views

Want to find a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies these inequalities

I want to find a function $f \in C^\infty([0,T])$ such that $$0 < L \leq f \leq M$$ $$f' \geq C \geq K_1 + K_2M$$ where $K_1$ and $K_2$ are fixed positive constants and are given. Is it possible ...
2
votes
1answer
380 views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
1
vote
2answers
63 views

How does Wolfram get from the first form to the second alternate form?

So, I was trying to compute an integral but I couldn't actually manage getting anywhere with it in its initial form. So, I inserted the function in Wolfram Alpha and I really got a nicer form (second ...
0
votes
0answers
60 views

Is there a method to list all periodic points for a funcion?

I search for a method that finds all periodic points of a given function e.g. $f(x)=x-x^2$ on its domain. You may explain some methods for a part of functions e.g. polynomials or $\mathcal{C}^k$ or ...
0
votes
1answer
24 views

Solution of quantile function

Find the quantile function of $$q=F(x)=[(1-\exp(-bx))^c]*[1+d-d*(1-\exp(-bx))^c]$$ , where $b, c$ are positive real and $-1<d<1$. Its answer is Any help/hint is most welcome.
0
votes
1answer
44 views

Finding the critical points of a function

What are the steps in finding the critical points of a function in general? Say for example, the function $$f(x, y) = 2x^3 + 11x^2 + 0.5y^2 - 2xy$$ I can't quite seem to understand the steps/method ...
0
votes
1answer
54 views

Degree of Equations [closed]

A) Which variables in the formula $V = \pi r^2 h$ would you need to set as a constant in order to generate: a linear equation? a quadratic equation? B) How should $r$ and $h$ be ...
0
votes
1answer
70 views

Guess a function that fits empirical data

This is my empirical data: Which function it looks like? I tried to guess (1) a dumped (exponential decaying) sinusoidal, but it does not oscillate after overshoot; (2) a sigmoid, but it oscillate ...
-1
votes
3answers
45 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$ [on hold]

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
0
votes
1answer
28 views

Slope of a function very much less than the function

I was working on a Cosmology problem and got stuck at this approximation used in a paper. Fundamentally the approximation is, $\frac{df(x)}{dx} \ll f(x)$. Now I can't understand how to imagine this ...
2
votes
3answers
336 views

Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation.

Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation $y = ax^3 + bx^2 + cx + d$. I have no clue how to solve this. This looks like ...
0
votes
1answer
23 views

Function with similar properties

Suppose I have a function $f$ and derive another function from it with similar properties. For example I have that my new function is zero when the other function is zero. I would still like to use ...
1
vote
2answers
35 views

Why is this inequation correct?

$$\sup |f(x)| -\inf |f(x)| \ge \sup f(x) -\inf f(x).$$ How can you show that it is indeed true?Sup is the lowest upper bound and inf is the greatest. lower bound
0
votes
2answers
279 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
1
vote
2answers
30 views

Redistributing Money in Monopoly

Is there a class of functions that satisfies the following properties? $\lim_{x \to -\infty}f(x)=-k, \lim_{x \to \infty}f(x)=k$ $f(x)<f(y) \Longleftrightarrow x<y $ ...
1
vote
2answers
53 views

Properties of bijections

If a bijection exists between set A={a1, a2, ...} and set B={b1, b2, ...} such that a1 maps to b1 and a2 maps to b2, etc., does this mean if we find a relationship R between a1 and b1 (i.e. f(b1) is ...
1
vote
1answer
52 views

Limit of average of real function

I need some hints regarding this exercice. if $f : [0, \infty)\rightarrow \mathbb{R}$ is a measurable function s.t $\lim_{x\rightarrow \infty} f(x) = a$, prove : \begin{align} \lim_{x\rightarrow ...
3
votes
3answers
40 views

Roots of real polynomial

$f(x)$ is a real polynomial. Show that $z=a+bi$ and $\bar z=a-bi$ have the same algebric multiplicity. I know that if $z=a+bi$ is a root of $f$ then $\bar z=a-bi$ is too, but don't know how to use ...
1
vote
2answers
83 views

Why are all non-polynomial functions are basically exponents?

There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. For example, $\log$ is simply inverse of ...
0
votes
1answer
38 views

Definition of the total variation of a function $g:\mathbb{R}\to\mathbb{R}$

if the total variation of a a real function $f:[a,b]\to \mathbb{R}$ over $\textbf{P}=\{a=t_0<t_1<...<t_m=b\}$ is $$ V^{a}_{b}(f)=\sup_{\textbf{P}}V(f,\textbf{P}) $$ where $$ ...
0
votes
1answer
47 views

Can we approximate $f(x) = \chi_{(0,\infty)}(x)$ by smooth monotone functions?

Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence? Preferably ...
1
vote
0answers
46 views

What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
0
votes
1answer
44 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...