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

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0
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2answers
23 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
0
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1answer
35 views

help with funky function definition

I've never encountered a function definition like this before and am wondering how you would go from this definition to finding out what features it has (y-intercept, even/oddness, min/max value, ...
0
votes
1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
1
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2answers
37 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
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0answers
29 views

Polynomials that represent a function

Let $D(x,n_1,\dots,n_k) \in \mathbb{Z}[x,n_1,\dots,n_k]$ be a polynomial. Every such polynomial represents a semi-decidable property of natural numbers by $$P(x) :\equiv (\exists n_1,\dots,n_k)\ ...
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1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
-2
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2answers
38 views

Graph of floor function [closed]

Please help me to draw the graph of $f(x)= \lfloor{x^2-1}\rfloor$. Please give me some tips.
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5answers
2k views

In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?

I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason why $\tan ...
1
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1answer
88 views

Inverse function of $x+\ln(x)$

How can I find the inverse function of $$f(x)=x+\ln(x).$$ This function has an inverse function (I can prove it) but I couldn't find it. Help please!
2
votes
1answer
68 views

I don't understand the mathematical definition of an inverse function

A function $f:X\rightarrow Y$ is called invertible if there exists a function $g:Y \rightarrow X$ such that: $y=f(x)\Leftrightarrow x = g(y)$ for all $x\in X $ and for all $y \in Y$ In ...
1
vote
3answers
201 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
1
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1answer
222 views

Function that is not differentiable at a point

I am looking for a continuous function to be used in fourier series graph that have the same value at both $-\pi$ and $\pi$ but has a very poor differentiability at a point. I have one: ...
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0answers
24 views

Keep a Function Positive via Mod

I have a function $F$ and I want it to remain positive--i.e., $$- F=F,\quad F=F $$ Would sticking a $\mod 2$ in front of $F$ do this? That is, because $$-1\mod 2=1\mod 2=1 $$ Then let ...
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
11 views

Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
0
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1answer
17 views

Function with single formula that have non zero value only in one small range

Is it possible to have one formula for a function $f(x)$ that will look like on this picture: Without cases for $x < x_0$, $x >= x_0\space and\space x <= x_1$,$x > x_1$. The formula ...
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0answers
18 views

Help in finding if the relation exist

I have this relation: $$ exp(-\phi \rho ) = \frac{h-1}{n-1} \psi(\phi, \rho ) $$ where $\phi \in [a,b] \subseteq \mathbb{R}^+ \backslash \{0\} $, $\rho \in [c,d] \subseteq \mathbb{R}^+ ...
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1answer
37 views

Why does $f(x) = x \cos 4x$ lie below the $x$-axis in ther interval $[\pi/4, \pi/3]$?

How can I explain why function $f(x)$ lies below the $x$-axis in the interval $\left[\dfrac{\pi} 4, \dfrac{\pi} 3\right]$ where $f(x) = x \cos 4x$?
1
vote
1answer
47 views

Using 4 step-rule $y = 2/ (4t - 3)^{2}$ [closed]

I tried solving it. My answer is $-4/16t^{2} + 48t + 18$, if your answer is different kindly show how is it done too thanks
2
votes
1answer
383 views

How to prove a function is periodic?

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?
14
votes
12answers
9k views

Piecewise functions: Got an example of a real world piecewise function?

Looking for something beyond a contrived textbook problem concerning jelly beans or equations that do not represent anything concrete. Not just a piecewise function for its own sake. Anyone?
0
votes
1answer
370 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
0
votes
1answer
22 views

Graphing functions

I am having problems understanding how to graph the product $fg$ when $f(x) = x$ and $g(x) = |x|$. Any help would be much appreciated!
1
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1answer
32 views

I need help proving this theorem (composition of functions)

This is the statement: If $f$ and $g$ are functions, the composition $g\circ f$ is a function with $$D(g\circ f)=\{x\in D(f):f(x)\in D(g)\}$$ $$R(g\circ f)=\{g(f(x)):x\in D(g\circ f)\}$$ The ...
2
votes
1answer
81 views

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto?

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto? I am not sure how to tell. Say $b\in N\times N$ this means the codomain is all the different combinations of the natural numbers. But ...
2
votes
2answers
24 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
3
votes
3answers
112 views

Finding if a function is onto?

Is the following function onto? It is a piece-wise function. Let the function $f:\mathbb{R}\rightarrow \mathbb{R}$ be $f(x)= \begin{cases} 2-x &, x\le 1 \\ \frac{1}{x} &, x>1 ...
10
votes
2answers
1k views

Why the name 'FACTORIAL'?

Factorial is defined as $n! = n(n-1)(n-2)\cdots 1$ But why mathematicians named this thing as FACTORIAL? Has it got something to do with factors?
2
votes
1answer
39 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
1
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1answer
33 views

Composition of functions which is one-to-one.

$f:Y\rightarrow Z$ and $g:X\rightarrow Y$ If $f\circ g$ is one-to-one then which of the following must be true? 1.$g\circ f$ is one-to-one. 2.g is one-to-one. 3.f is one-to-one. 4.g is onto.
1
vote
1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
0
votes
1answer
39 views

Derivation of Dirac-Delta with complicated argument $\delta(f(x))$

Recently I learned how to deal with the Derivative of a shifted Dirac-delta. Now I want to go a step further, but are not sure about the solution. Is there a simple way to rewrite terms like this ...
0
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0answers
37 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
1
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3answers
105 views

Is $4x^2-4x+2$ surjective?

Determine whether the function $f_4:\mathbb{R^+}\rightarrow \{x \in \mathbb{R^+} x \ge 1\}$ given by $f_4(x)=4x^2-4x+2$ is injective, surjective or bijective. I will just show parts of the solution I ...
2
votes
3answers
140 views

How to find know if function is onto?

How do you figue out whether this function is onto? $\mathbb{Z}_3\rightarrow \mathbb{Z}_6:f(x)=2x$ Onto is of course is for all the element b in the codomain there exist an element a in the domain ...
3
votes
2answers
59 views

Function for this game movement graph?

This may be too easy for you, but here goes: I'm creating a kind of ski slalom game where I want the horizontal speed/direction to follow the attached graph. X is time, Y is horizontal speed. Positive ...
0
votes
0answers
13 views

Functions to fill spheres at certain points in a cylindrical volume?

I know how to find how many spheres of a given radius can fit into a cylinder. But, I also want 2 know how to fill them throughout points in the cylinder. To give an example I have spheres of a ...
1
vote
3answers
107 views

Why isn't an injection an iif?

Suppose that $f:X \rightarrow Y$ is a function. Then an injection can be defined as: $\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$ Why isn't it defined instead as follows: $\forall ...
0
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0answers
28 views

Reversible smoothing of a two dimensional function (or an image)

Smoothing of an image, or a two dimensional function is quite easy, there are many methods to achieve it, using average of near elements. But how to make it reversible? Maybe DCT (discrete cosine ...
0
votes
1answer
28 views

Need function for 2D sigmoid-shaped monotonic Surface

I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ ...
-1
votes
1answer
18 views

Rewriting a quadratic function

i have to find domain of this function $f(x) = \log(10+3x-x^2)$ can i rewrite this as $\log(x^2-3x-10)$? I found the domain but is not the same for 2 forms of function.
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votes
4answers
35 views

Showing that $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f(x)=[3x]$ is function?

How can I show this is a function? $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f([x]_4)=[3x]_2$ where $[x]_n$ is the equivalence class of $x\mod n$. I think it is a function because I cannot ...
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votes
4answers
53 views

Solving a problem using the definition of limit [closed]

How can I solve this using the definition of limit? Prove using the definition of limit that: $$\lim_{x\to 1} (x²-4x)=-3$$ How can I approach this? EDIT: OH my god! Thanks @adam! Maybe you ...
3
votes
2answers
53 views

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$.

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. EDIT: Actually, this identity should hold even if $f$ is not one-to-one (injective), right? ...
6
votes
1answer
69 views

What am I doing wrong in this algebra excercise?

This is my first question here, so please forgive me if the format etc. are not quite right. I've been attacking an algebra question, and my workings are below. There's a mistake somewhere (I don't ...
1
vote
3answers
56 views

True or False Question About Functions [closed]

If $f(1)>0$ and $f(3)<0$, then there exists a number $c$ between $1$ and $3$ such that $f(c)=0$. I'm not sure how to solve this question. Thanks in advanced!
0
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1answer
30 views

What is the domain? [closed]

What is the domain of the function $f(x) = \sqrt{4 x + 37}$? I am not sure where to get started with this problem. Thanks in advance!
0
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0answers
30 views

Find domain of a function

Function is : $f(x) =\sqrt{ 16 - x^2}$ First i change side $x^2-16\ge0$, then $(x+4)(x-4)\ge 0\implies x=-4,x=4$. I found that domain is:$(-\infty,-4] \cup [4,+\infty)$ but the problem is: this ...
2
votes
3answers
38 views

Is the inverse of this function unique

Let $f$ be a function from any set(Say $K$) to any set (say $P$) Now: $f(x)=2x+1$ My question:Is it necessary that the inverse of the function is $\frac{x-1}{2}$? This is a problem given in my ...
0
votes
0answers
30 views

Sketching the graph of a function with three real roots

I need to solve the following question: Sketch a graph of a function $f(x)$, continuous in all $x \in \Bbb R$, knowing that $f$ has three real roots, that $\lim_{x\to+\infty} \left[f(x)-\frac ...