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

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2
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0answers
18 views

Real, bounded, convex function [duplicate]

I need to show that a real, bounded, convex function whose domain is $\mathbb{R}$ is constant. I don't know where to start. Thanks in advance!
0
votes
0answers
23 views

Notation for partial function set.

There is a standard notation for the set of all functions between S and T, namely T^S. Is there a similar notation for the set of all partial functions between S and T?
1
vote
1answer
31 views

$f$ is an even function defined on $(-5,5)$

If $f$ is an even function defined on $(-5,5)$ on the interval, then find four real values of x satisfying $f(x)=f(\frac{x+1}{x+2})$. My book gives the answer as $\frac{\pm3 ...
0
votes
2answers
27 views

How to solve this question subset

Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample. For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then ...
2
votes
1answer
58 views

Sketch the Graph. $S(x) = \frac{1}{2} (x-1)^3 +4$

The question is: Sketch the graph of the the function by transforming the graph of an appropriate function of the form $y = x^n$. Indicate all x and y intercepts on the graph. I am really trying to ...
1
vote
3answers
33 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
0
votes
1answer
22 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
vote
2answers
65 views

Find the range of a function

How to find the range of the following trigonometric function $\sin^2x-5\sin x-6$. Can some one help me out. Thank you
0
votes
1answer
39 views

Confusion surrounding functions

Hey there Mathematics, Slightly confused over some of the things in my quiz and was wondering if I could get an explanation: I thought with the first question that it's just one-to-one from X to Y ...
0
votes
1answer
10 views

Class of functions with constrains on derivative

In an article I am reading, the author considers a class of functions: $$ \{T \in C^1(\Bbb R^+)\text{ strictly increasing; } \ \ x\le T(x)\le xT'(x) \} $$ Can you give me a generic example of such a ...
2
votes
2answers
22 views

maple evaluate function of variables containing x

Well this is basically the problem. I want to evaluate a function but the variable x is defined inside a, so for some reason maple doesn't substitute 7 for x.
0
votes
1answer
24 views

State the range of the function below.

Sketch the graph of $f:x \mapsto -4x + 5$ , $x<2$ and state the range. I got the graph, but can't state the range...how to find them?
0
votes
2answers
25 views

Prove a function is surjective and injective

Let $f\colon X \to Y$ be any function and let $g$ be the following function: $g\colon P(X) \to P(Y)$, $g(S) = \{f(s) : s\in S\}$ Prove that: 1) If $f$ is injective then $g$ is injective. 2) If ...
3
votes
2answers
43 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
0
votes
0answers
6 views

Check sum of functions does not exceed value.

I have functions f(x) and g(x) defined as always being 0 except over a defined interval. (Ie: between -10 and 10) I would like to know what the proper syntax is to state that there should be no value ...
1
vote
2answers
29 views

Horizontal Asymptote & Range

Suppose y = c is a horizontal asymptote of a function y = f(x). Is it true that the number c does not belong to the range of f(x)?
1
vote
1answer
41 views

Problematic integral

Yesterday I started to solve the integral of $\frac{1}{x^5+1}$ in $dx$ so I started in this way: $$ \int \frac{1}{x^5+1} dx = \int \frac{1+x^5-x^5}{x^5+1}dx = \int \frac{x^5+1}{x^5+1}dx - \int ...
7
votes
3answers
233 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
2
votes
4answers
75 views

Find the number of elements in the range$ f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3$.

Find the number of elements in the range $f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3.$ I cant understand...It will go very long if i keep breaking them into small intervals .
1
vote
2answers
42 views

Prove that if $f(x) = Ax^2 + Bx + C$ is an integer whenever $x$ is an integer, then $2A$, $A+B$ and $C$ are also integers.

Prove that if $f(x) = Ax^2 + Bx + C$ is an integer whenever $x$ is an integer, then $2A$, $A+B$ and $C$ are also integers. I've tried a lot to do it, but can't get it exactly right.
0
votes
0answers
23 views

Find a bijective function between the following two sets

Define a bijective function g g : P(Z) ---> P(N) Where P is the power set, Z is the sets of integers and N is the set of natural numbers (which include 0). What I find difficult is that I don´t ...
0
votes
2answers
70 views

How to rigorously find range of a function?

What is a rigorous method/mechanism to find the range of a function $f(x)$? Is it acceptable to find $f^{-1}(x)$ and then its domain? I understand that $f^{-1}(x)$ does not always exist (at least not ...
2
votes
0answers
65 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
1
vote
1answer
79 views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ ...
1
vote
0answers
30 views

Elementary embeddings, elementary substructures,category of sets

I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
1
vote
1answer
55 views

Finding the integral of a radical function?

I had to create a problem in Calculus, and since I'm terrible at finding the limits of a definite integral, I decided it would be good to practice with that. So I created this: Two waffles are ...
0
votes
0answers
16 views

Move Hill equation curve horizontally without changing its shape?

I have a normal Hill function of: $y = \dfrac{x^\lambda}{h^\lambda + x^\lambda}$; where $\lambda$ is Hill coefficient, and $h$ indicates the infection point. I am concerning if we could add another ...
0
votes
0answers
34 views

Question about an exponential funtion

Now we have a function: $f(x)=e^x, x\in \mathbb R$ Question: 1) Assume that $x>0$, discuss the number of the intersects between $f(x)$ and $y=mx^2,m>0$ under different situation. 2)Assume ...
0
votes
1answer
16 views

How can I show that a “function” assigns exactly one value to each x in its specified domain?

I am teaching myself using a book, Mathematical Analysis, authored by Bernd S. W. Schröder. In Exercise 1-30 (b) (i) at page 16, it says that Function $f:Q\rightarrow \tilde{R}$ is described in some ...
0
votes
0answers
25 views

Function plotting

I have a function $f(x)=\binom{N}{K} \ln(1-F(x)), x \geq 0$, where $F(x)$ is a cumulative distribution function. Then, $\ln(1-F(x))$ is negative for various values of $x$ as $F(x) \geq 0$. Also, ...
0
votes
0answers
21 views

Draw arrow diagram to show the following function.

Draw arrow diagram with two parallel lines to show the function $f:x \mapsto 3 - 2x^2$. Let the domain be the set of integers and draw six arrows for the function. How to draw it?
3
votes
2answers
84 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
1
vote
2answers
44 views

Aren't $ f’(xy) $ and $ f’(x/y)$ ambiguous notations? [Stewart P890 14.3.50b, c]

The answers so far have uncloaked a deeper problem of mine: I brook that setting $t = xy$ transforms $f(xy)$ into $f(t)$, but didn't we start with $x, y$ which are 2 separate independent ...
0
votes
0answers
9 views

An estimate concerning certain twice differentiable function

Let $f:(-1,1)\to \mathbb R^+ \cup ${$0$} be a twice differentiable function such that $f(0)=1$ and $f'(x) \leq 0 , \forall x \geq 0$ ; $f(x) \geq f''(x) , \forall x\geq0 $ , then is it true that ...
0
votes
1answer
31 views

Hi guys, can anyone help with this recurrence relation problem?

I'm going through practice questions for my exams but this question has left me confused: The Bessel functions of integer order, Jn(x), are described by the generating function: Derive the ...
0
votes
0answers
22 views

`x` increases by 10 everytime`y` occurs`i` times. How do I calculate the sum of `i`?

Can anyone tell me how to calculate this? x increases by 10 everytime y occurs i times. How ...
0
votes
1answer
24 views

Regression function - conditional mean

I am trying to understand the statistical fundamentals behind linear regression, and i have never been able to intuitively understand the following; really would appreciate if someone could give an ...
0
votes
1answer
35 views

Union of preimages [duplicate]

Given $f:X\rightarrow Y$ as a function, the image of $x$ if $f(x)$. The preimage of $y$ is $f^{-1}(y)=\{x\ |\ f(x)=y\}$, with the symbol PreIm$(Y)$ Given the definition, could you prove the following ...
0
votes
0answers
27 views

Test Perfect Square of any Function

consider i have equation like this: f(x) = 5X^2 + 4 my question is, how can i check f(X) whether is a perfect square or not. But, suppose we can't squaring X because of computer data type ...
1
vote
2answers
22 views

Help with Discrete Math Functions and Bijections

I have trouble with the following problem: Prove that the function $f(x)=x^2-2x+3$, with domain $x\in (-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: I tried to first prove ...
-1
votes
0answers
19 views

Google Doc Algorithm To Calculate Incrementing Sum?

Ok I'm trying to calculate the total XP used for each stat but can't seem to figure out the algorithm. Stat Value is editable and changes to calcs in the Next Cost and XP Used. CostX is how much a ...
-1
votes
1answer
66 views

Equation $x^3+ax^2+bx+26=0$. Find the sum of the absolute values of the possible values of a.

I tried it by putting the sum and product formula into work for this equation... All roots are integer.
0
votes
1answer
27 views

Help with Functions in Discrete Mathematics

I am having trouble solving this problem: let $p$ be a positive prime number and let $f:Z_p -> Z_p$ be defined as $f([x])=[x^2]$. Show that $f$ is a function. Give examples of how it is not ...
-1
votes
1answer
18 views

Continuity of a Piecewise Function for Rationals and Irrationals [closed]

Consider the function that maps each irrational number to 0 and each rational number $n/k$ to $1/k$ where $n$ and $k$ are in lowest terms. Where is the function continuous?
1
vote
0answers
23 views

Deriving recurrence relations, very stuck!

Going through past papers for my exams and cannot figure this one out, does anyone know how to do these? The Bessel functions of integer order, Jn(x), are described by a generating function of the ...
0
votes
3answers
62 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...
0
votes
0answers
13 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
0
votes
1answer
31 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
7
votes
5answers
167 views

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
2
votes
1answer
37 views

Examples of contractions between functional spaces

Define $\mathcal{F}$ as the following set of continuous functions: $$ \mathcal{F} := \left\{ f: \mathbb{R} \rightarrow \mathbb{R}^n \mid f(\cdot) \ \text{contin.}, \ f(x) \in K(x) \subset ...