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

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0
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1answer
21 views

Finding a population Function

I have been given the population of the USA from 1790 - 1980 (increasing in intervals of 10) and I am asked to solve this differential equation. Using t as time in yearrs P as size of population at ...
0
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2answers
42 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
0
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0answers
30 views

Function's graph sketch

$f(x) = x+9$ $g(x) = 2x-3$ I need to draw the sketch of $\min(f,g)$ and $\max(f,g)$. I tried: http://sketchtoy.com/62375984 The yellow is the min and the blue is max. The lines should be ...
0
votes
2answers
29 views

Getting a diverse set of three numbers from two numbers

I'm using this information to build an interface to pick a color, but I feel that this question is purely math-related. Please correct me if this is the wrong StackExchange site for this. I am making ...
0
votes
2answers
42 views

Geometric meaning of results obtained in (a) and (b)

The task: Plot the function $\sqrt{1-x^2}$. What does it look like? What is the geometric meaning of the results you obtained in (a) and (b)? Can anybody help me with geometric mean? I can't ...
1
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2answers
37 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
0
votes
1answer
22 views

How to make clear a letter is a function?

How should I make clear that a symbol is a function? Usually a function is denoted by the letter $f$ or $g$, or is directly applied to arguments (e.g. $c(x,y)$) or is implied to be a function by an ...
0
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2answers
49 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
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2answers
42 views

Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this. I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one ...
6
votes
2answers
68 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
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3answers
28 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
0
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0answers
13 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
0
votes
1answer
33 views

Figuring the function $f(x)$ from given information

Here is the given information in my question, So, what my question inform is that there is a cubic polynomial function (i.e $f(x)$) which has local maxima at $x=-1$. While that for $f'(x)$, it's ...
0
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0answers
32 views

Iterative function eventually reaching identity

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that ...
0
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2answers
19 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
0
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1answer
29 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
1
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1answer
40 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
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0answers
26 views

Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
-1
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0answers
24 views

Need some help with a function on a graph.

I need to make a function that starts pretty fast going up but then slows down but still goes up. Thanks in advance.
2
votes
1answer
48 views

Proof that there is a bijection, if there are injective maps in both directions

Let $A$ and $B$ be two sets. Let $f:A\to B$ be injective such that $Im(f) \subsetneq B$. Let $g:B\to A$ be injective such that $Im(g) \subsetneq A$. Obviously $A$ and $B$ are not finite sets. Can ...
1
vote
1answer
19 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
2
votes
4answers
34 views

Heaviside Unit Step Function

Convert to heaviside function: $$f(t) = \begin{cases}e^t ,& 0 \leq t \leq 1 \\0 ,& t > 1\end{cases}$$ My attempt: $f(t) = U(t) e^t - U(t-1) e^t $ I think my solution is not right because ...
8
votes
1answer
611 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
1
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2answers
41 views

Help with function $f_r(x^q)=q^rx^{q-1}$

Let $r,q$ be a positive integers. I am looking for a function $f_r(x^q)$ such that it is satisfied $$ f_r (x^q)=q^r x^{q-1}$$ (without explicit dependence on $q$ of course, and for $r>1$). I ...
0
votes
1answer
47 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
1
vote
2answers
98 views

Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
0
votes
1answer
41 views

How to prove that a given map is an injection?

Let $g:\mathbb{N_{m_1-1}}\rightarrow \mathbb{N}_{m_1}$, where: $$g(i) = \left\{ \begin{align} i & \text {, for } i<i_0 \\ i+1 & \text{, for } i \ge i_0 \end{align}\right.$$ and $i_0 ...
2
votes
1answer
22 views

Surjectiveness of standard-normal c.d.f. [on hold]

Let $\phi:\mathbb R \to (0,1)$ be a function defined as $\phi(y)=\int_{-\infty}^y\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac {x^2}{2}}dx , \forall y\in \mathbb R$ , then is it true that $\phi$ is surjective ? If ...
0
votes
1answer
21 views

composing one function as a function of another function

I have two functions: $f_1=\sum_i x_iy_i$ and $f_2=\sum_i x_iy_i^2$, where $x_i$s and $y_i$s are positive and smaller than $1$. I want to write one of them as a function of the other (for example ...
1
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0answers
30 views

Why is there information loss here?

I posted this a while ago, but realized that actually this works, but produces information loss. Here is the process: Let's try to reduce a number $x$. We will be using base $\beta$ for this process. ...
2
votes
1answer
32 views

Composition and Limits

Suppose that $f$ is a continuously differentiable function with $\lim_{x \rightarrow \infty} f(x)=k$ and $g$ is a Lipschitz continuous function. Prove that $\lim_{x \rightarrow \infty} ...
0
votes
1answer
22 views

An example of a twice continuously differentiable and bounded function.

Find an example of a twice continuously differentiable and bounded function $f:\Bbb R \rightarrow\Bbb R$ such that $\lim\limits_{x \rightarrow \infty} f(x)$ exists, but $\lim\limits_{x\rightarrow ...
1
vote
2answers
39 views

Fixed points of a certain type of functions with intermediate value property

Let $f: \mathbb R\to \mathbb R$ be a function, having intermediate value property, such that $f(f(x))=x , \forall x \in \mathbb R$, then is it true that either the set of fixed points of $f$ is ...
0
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0answers
16 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
2
votes
3answers
34 views

How do I find the domain of this function

I would like to know which operations i have to do to get the domain of this function: $$y=\sqrt{\frac{1}{x}-1}$$ I have researched and the solution of the inequality $\frac{1}{x}-1 \geq 0$ is ...
1
vote
1answer
10 views

Divisibilty of a functional equation

I found this question in a mathematical problems book: Let $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$. Prove that $f(x^n)$ is divisible by $x^n-1.$ Can anybody help me?
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0answers
23 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
0
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0answers
18 views

Is the following subset a vector subspace of F(R,R). How to prove that it is a subspace.

Is the following subset a vector subspace of F(R, R)? The set of functions f : R → R such that f(x + 2) = f(x) for all x ∈ R I know this is obviously a subspace, but the mark scheme didn't ...
2
votes
0answers
17 views

Boundedness of a certain function defined on a closed bounded real interval

Let $I:=[a,b]$ be a closed bounded real interval , $f: I \to \mathbb R$ be a function such that for every $x \in I$ , $\exists \delta_x>0$ such that $f(x)$ is bounded $ \forall x \in ...
0
votes
1answer
20 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
0
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1answer
41 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
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0answers
21 views

How to check the availability of particular function [closed]

I want to check the availability of particular function that returns the value of results in binary like 0 or 1....How to check these function using mathematics...
2
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2answers
74 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
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1answer
25 views

parameterize the following functions [closed]

Help determining the parameterized solution of the following functions $$a) { \left( x-2 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 }=4\quad if\quad 1\le y\le 3$$ $$b) \frac { { \left( x+3 \right) ...
1
vote
2answers
50 views

Graph a function

I have a question, I have a function: $$f(x) = \frac{-x^2-10x}{2}$$ I'm really confused how to replace the x. So, what would be the points in $y$ if $x$ were: $-4, -3, -2, -1, 0, 1, 2, 3, 4$?
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0answers
29 views

Need a parabolic equation using two points and the slope at those points.

Can someone give me a function to solve any parabolic equation that has two known points with known slopes? Thanks much. Example: Point 1: (x1, y1), slope a Point 2: (x2, y2), slope b
1
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0answers
31 views

Math Formula For A Loyalty System

Basically, I help someone manage a stream on twitch.tv. She uses a program that rewards the viewers with a virtual currency. For every 30 minutes they watch they get 1 point. Also, they get a 1 ...
0
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0answers
31 views

Exercise about vector functions of real variable

Parametrize in a clockwise direction by means of a continuous vector function into pieces, starting at the point $(1,1)$ the following curve. (Sorry for my bad English) $$c:\begin{cases} { \left( ...
1
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1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...