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

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3
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1answer
530 views

Nested Division in the Ceiling Function

During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that: $$ \left\lceil{\frac{n}{2^i}}\right \rceil= ...
0
votes
1answer
122 views

Conditions for which two functions, one containing an integral, are equal

Define two functions: ${\beta _1}(\tau ,\omega ) = \exp \left[ {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$ ${\beta _2}(\tau ,\omega ) = \exp \left[ {\frac{{\omega \tau ...
0
votes
3answers
173 views

Find all function $f$ such that $f(x)+f(\frac1x)=\frac1a; a$ is constant

Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?
1
vote
1answer
304 views

Probability - Finding the Cumulative Distribution Function

If $X$ is uniform on $[0, 1]$, and $Y$ is a discrete random variable which is either $1$ or $2$ with probability $\frac12$ each, find the cumulative distribution function of the product $XY$. I know ...
8
votes
1answer
84 views

Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?

This is my first question and I hope this question is not too brief to be acceptable: There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
1
vote
1answer
52 views

Is it a known equation?

$$\int |f(x)|\, dx = \int f(x) \, dx\cdot\frac{1}{f(x)}\cdot|f(x)|$$ with $|x|$ is the absolute value of $x$. The equation is nothing new, I guess? Regards
6
votes
1answer
335 views

Infinitely times differentiable function

Let $f$ belong to $ C^{\infty}[0,1]$ and for each $x \in [0,1]$ there exists $n \in \mathbb{N}$ so that $f^{(n)}(x)=0$. Prove that $f$ is a polynomial in $[0,1]$. I am trying to use Baire Category ...
1
vote
1answer
234 views

Minimizing a linear combination of convex functions

Suppose a series of convex functions $f_1(x), f_2(x), f_3(x), ...$ is given (also, expressions for their derivatives $\nabla f_1, \nabla f_2,...$ are known). Now, suppose a function $g(x)$ is a linear ...
1
vote
2answers
55 views

Would cartesian product be the best approach for this

Not sure on how to migrate a question yet but over on SO someone said I might get better results here. Also please retag as I'm not allowed to create new and might not know the best tagging. Link to ...
0
votes
1answer
613 views

show that: $f$ is injective $\iff$ there exists a $g: Y\rightarrow X$ such that $g \circ f = idX$

** proof under construction - will post when done and more or less confident it's true. ** also please easy with the downgrades.. i don't understand why i'm getting them. what is meant by show ...
3
votes
0answers
165 views

absolute value integrated (2)

Part 1 of the question Hey again, I hope that I was able to solve my problem. This is my solution with the example of Part 1 of the question. Is that correct or did I forget anything? $$ \int ...
2
votes
3answers
601 views

absolute value integrated

For example I have the function $f(x) = |x^2-1| = \sqrt{(x^2-1)^2}$ $\int \sqrt{(x^2-1)^2}dx = \frac{x(x^2-3)\sqrt{(x^2-1)^2)}}{3 (x^2-1)}+constant$ But plotted it looks like this: Plot 1. There are ...
0
votes
1answer
177 views

Inverse mapping induced by mapping

In Folland, there is a statement as follows: Any mapping $f: X \to Y$ between two sets induces a mapping $f^{-1}: \mathcal{P}(Y) \to \mathcal{P}({X})$ (these are power sets) defined by $f^{-1}(E) ...
0
votes
2answers
85 views

Showing the Uniqueness of a Function

The following question is distantly related to the Fundamental Homomorphism Theorem: $\fbox{Let}$ $f:A \rightarrow B$ $g_1:B \rightarrow C$ $g_2:B \rightarrow C$ and finally assume $g_1 \circ f = ...
3
votes
2answers
776 views

Extreme points of unit ball in $C(X)$

Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuos functions in sup-norm. I read in Douglas' Banach algebra techniques in operator theory that the followings are equivalent: ...
5
votes
3answers
489 views

Physical meaning behind Frequency domain?

I understand its usage and why is it important because It transforms differential equations to algebraic ones.. But I can't get the physical meaning of the new form of the equation and the meaning of ...
4
votes
1answer
1k views

Finding out a rational equation via a graph

I need to be able to find an equation from this graph So far I have this graph with the equation $-1/((x-3)^3)$ I can see from the desired graph that there is no horizontal asymptote, compared ...
1
vote
0answers
237 views

Supremum or infimum of a function

Does there exist an elementary function on some subset $I$ of $R$ such that you can prove that $A = \{\sup(f(x)): x \in I\}$ exists, but you cannot find the value of $A$ ? If the answer is "no", then ...
1
vote
0answers
85 views

Ramanujan phi-Function

by chance I found http://mathworld.wolfram.com/RamanujanPhi-Function.html and wanted to understand the definition, so I tried to prove the conclusion on my own. I wrote the things I saw as given in ...
19
votes
3answers
1k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
2
votes
3answers
99 views

Understanding simplification of an algebra problem

I have the problem $x^3 - 2x^2$. My book tells me that this problem is simplified to $x^3 (1 -(\frac{2}{x}))$. How does that work? This step of my book I am in about the "end behavior" of trying to ...
3
votes
3answers
400 views

Prove $f(S \cup T) = f(S) \cup f(T)$

$f(S \cup T) = f(S) \cup f(T)$ f(S) encompasses all x that is in S f(T) encompasses all x that is in T thus the domain being the same, both the LHS and RHS map to the same ys, since the function ...
1
vote
0answers
26 views

Series function help

I want to find a function such that $$ \sum_{0<j<n/k } f(kj)=1 $$ Where the sum j is taken over the natural numbers, And the series is satisfied for all integers k and n, I was thinking of ...
0
votes
2answers
187 views

Power functions and parabola issue

With the function f(x)=x^2 we get a graph like so... The rule for power functions, that I've been told, is the larger the power gets, the closer the line will touch the x-axis. Example for ...
2
votes
0answers
117 views

uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
0
votes
0answers
90 views

No approach coming in mind

I found this question from a book ..... I dont know the solution neither i can now find the book ) Let the function $f_n(.....f_3(f_2(f_1(x))))$ be an increasing function. If r of the individual ...
0
votes
1answer
239 views

Continuity of $\sqrt{x}$ at a point

I have some problem to understand the definition of a continuous function in a point. I have $f(x) = \sqrt{x}$ and I want to check the continuity of the function above in the point $x_0 = 0$ or for ...
2
votes
2answers
2k views

Construct a polynomial function with the given graph

Where does one begin? I can see that the zeros are -5, -3, 0, and 4? Is that correct so far?
1
vote
3answers
348 views

Binary Sequences

Let $B_n$ = $\mathcal{P}(\{1, 2, \dots, n\})$. The set $\{0,1\}^n = \{a_1, a_2, ... , a_n : a_i \in \{0,1\}\}$ is called the length of binary sequences of length $n$. I want to verify and work on ...
0
votes
1answer
95 views

Inverse of a function

From my text book it says that $f(x)= x^3$ and $f^{-1}(x) = \sqrt[3]{x}$ , which I totally agree with. why does $f(x)= \frac 1 {x-1}$ and $f^{-1}(x)= \frac 1 {x + 1}$ and not equal $f^{-1}(x)= \frac ...
1
vote
2answers
136 views

Composition function

If $f:S \to T$ and $g:T\to U$ are functions how can I prove that if $(g o f )$ is one-to-one, so is $f$, and find an example where $(g o f )$ is one-to-one but $g$ is not one-to-one.
2
votes
2answers
230 views

proving a simple function is bijective

This is more of a "How to write" question than a "help me solve" one, sorry if these are unaccepted/closed, let me know and I won't open anymore like this. I need to prove that $A:=\{x\in ...
14
votes
11answers
1k views

If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?

If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$? Thank you.
2
votes
2answers
195 views

What are the strategies I can use to prove $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$?

$f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$ I think I have to show that the LHS is a subset of the RHS and the RHS is a subset of the LHS, but I don't know how to do this exactly.
0
votes
2answers
87 views

What are the different characteristics of a composite function?

Suppose the function $g$ and $f$ are one-to-one. Is $f \circ g$ one-to-one? Suppose $f \circ g$ is one-to-one, are the function $g$ and $f$ one-to-one? Suppose $f \circ g$ is onto, are the function ...
0
votes
3answers
186 views

Is this a valid proof of $f(S \cap T) \subseteq f(S) \cap f(T)$?

$f(S \cap T) \subseteq f(S) \cap f(T)$ Suppose there is a $x$ that is in $S$, but not in $T$, then there is a value $y$ such that $f(x) = x$, that is in $f(S)$, but not in $f(S \cap T)$. Suppose ...
0
votes
1answer
40 views

Need this Function Explained

I'm working on a personal project and I did a search for an integer sequence on the site listed below. I found the sequence I was looking for with a short description of the function defining the ...
1
vote
2answers
64 views

What does this function calculate?

Can anyone confirm to me what this function calculates? Function m(t); p=9t; q=p+32; r = q/5; Return r; End function Ok so inputting numbers from $0-9$ (in ...
2
votes
3answers
4k views

What are the different ways to prove a function is onto

you take $f(x)$ and isolate $x$? For instance $f(x) = x^2 = \sqrt{y} = x$ Set A = all rational numbers Set B = all rational numbers The function is not onto because $\sqrt{y}$ is an irrational ...
1
vote
1answer
33 views

Limit of Function that Satisfies Simple Property [duplicate]

Possible Duplicate: Limit of a continuous function Let $f$ be a function that is continuous and real on $[0, \infty]$ such that $\lim_{n \to \infty} f(na) =0$ for all $a>0$. What can be ...
1
vote
1answer
55 views

General solvability at the stationary condition

Suppose a convex quadratic function $f(x)$ is given. To find a minimum of such function, one sets its derivative so zero, and solves for $x$. For instance, suppose that the result of differentiation ...
1
vote
0answers
221 views

Is this function bijective, surjective or injective?

Is $f:\mathbb{R}^2\to\mathbb{R}^2$ given by $f(x,y)=(2x+y,3\lfloor y\rfloor-x)$ bijective? if not is it injective or surjective?
2
votes
4answers
261 views

How to calculate the maximum value of: $\frac{25x}{x^2+1600x+640000}$?

Wolfram says it's 800, but how to calculate it? $$ \frac{25x}{x^2+1600x+640000} $$
1
vote
2answers
450 views

How many solutions does $\cos(97x)=x$ have?

How many solutions does $\cos(97x)=x$ have? I have plot the function. However I don't know how to solve the problem without computer. Can anyone give a fast solution without a computer?
1
vote
1answer
77 views

Recursive function

For a sequence $a$, $a_1=2$, $$a_n=\frac{n-1}{a_{n-1}}+n-1$$Express $a_n$ in terms of $n$. I tried keep expanding, got many level of fraction until $n=1$, but I still can't see the pattern. ...
0
votes
1answer
107 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
1
vote
6answers
107 views

limit of convergent series

What is the limit of $U_{n+1} = \dfrac{2U_n + 3}{U_n + 2}$ and $U_0 = 1$? I need the detail, and another way than using the solution of $f(x)=x$, as $f(x) = \frac{2x+3}{x+2}$ because I can't show ...
0
votes
2answers
58 views

Cases when the Intermediate Value Theorem is true - Part 2

I previously asked this question and was told that an answer is certainly possible but I am still looking for an example. The question was for cases when the intermediate value theorem is true and a ...
1
vote
7answers
364 views

Is there any function where $f \circ f = f$ but $f(0) = 1$

Other than the identity function, is there any function where $f \circ f = f$? $f(0)$ also has to return 1. It must has something to do with the exponent 0 to a some coefficient... Anyone could give ...
2
votes
1answer
63 views

Composition of two functions in $\mathbb{Z^2}\to \mathbb{Z^2}$

I need to find the composition of a function and its inverse so I have the identity function in return. My problem is that I don't seem to undestand how to proceed algebraically. I have a function ...