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

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0
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0answers
15 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
2answers
24 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
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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?
0
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0answers
18 views

Property of linear growth function [on hold]

Prove: Suppose function $f(X)$ has linear growth and there exists an $X^*$ such that $f(X)$ is monotonic for $|X|>X^*$. Then it is possible to define a function $g$ such that $g$ a rotation of $f$ ...
0
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0answers
22 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
11 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 ...
1
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0answers
15 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
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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
votes
1answer
36 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
20 views

How to check the availability of particular function [on hold]

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
67 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 ...
-2
votes
1answer
21 views

parameterize the following functions [on hold]

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
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2answers
46 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
28 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
29 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
30 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: ...
0
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0answers
19 views

Interesting properties of functions and sets that depends on dimension of space.

For $n=1$ (or $m=1$), we have some basic properties of functions and sets that are not valid (or not necessarily valid) for $n\neq 1$ (or $m\neq 1$). For exemple: Calculus. Let $[a,b]$ be a closed ...
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 ...
1
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2answers
30 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
0
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1answer
32 views

How to deduce number of unordered distinct pairs using set operations and bijections

In (b) of the example, we are ask to calculate the number of ordered pairs of distinct positive integers. I like the first method's answer (using bijections, set operations) because it clearly shows ...
0
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0answers
7 views

Assessment of function $f(T^*)=1-\frac{d\phi(T^*)}{dT}e^{-\phi(T^*)}\int\limits_{T0}^{T^*}e^{\phi(T)}dT$

Is it possible to assess the following function: $f(T^*)=1-\frac{d\phi(T^*)}{dT}e^{-\phi(T^*)}\int\limits_{T_0}^{T^*}e^{\phi(T)}dT$ where $\phi(T)$ is an positive monotonic, differentiable, bounded ...
0
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0answers
41 views

Two definitions of functions

In literature on logic and set theory, there seem to be two different definitions of functions, one more general than the other. First of all, a function $f\colon X\to Y$ consists of three element ...
0
votes
3answers
99 views

What is the $\lor$ symbol?

In researching the consensus algorithm, I came upon the consensus theorem: How does the $\lor$ symbol function?
0
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5answers
35 views

Smallest value of function on a line

Problem : If the point $(\alpha, \beta)$ lies on the line $2x+3y=6$, the smallest value of $\alpha^2+\beta^2$ is (a) $36/13$ (b) $6\sqrt{13}/13$ (c) $6$ (d) $13$ Solution : Since ...
2
votes
1answer
37 views

Relations and functions with valence 0

From http://en.wikipedia.org/wiki/First-order_logic#Non-logical_symbols Relations of valence 0 can be identified with propositional variables. For example, P, which can stand for any ...
16
votes
3answers
1k views

I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
0
votes
1answer
20 views

Express function counting number of elements in subsets

I wish to express a function $freq(x)$ as an equation but I have no clue how to properly do this. Basically I have the following: Let $a_i \subset A$ be one of many subsets of A. Each subset $a_i$ ...
2
votes
4answers
111 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
1
vote
2answers
20 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
1
vote
1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
3
votes
1answer
33 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
0
votes
0answers
45 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
2
votes
2answers
49 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
0
votes
2answers
66 views

As we call “1/x” inverse x, how we name “ -1/x”?

Just as sin(x) represents some f(x), where the f(x) = opposite/hypotenuse and it seems redundant to create another function csc(x) = the inverse of sin(x) As we call "1/f(x)" inverse f(x), then how ...
-1
votes
2answers
86 views

Showing that $A\rightarrowtail A \times \{x\}$ is a bijection

$A\rightarrowtail A \times \{x\}$ where $A$ is any set and $\{x\}$ is an arbitrary one-object set. How would I show the following is a bijection ( one to one and onto)? I know if I turn it into a ...
2
votes
2answers
33 views

Composition of injections (proof)

I'm trying to prove that composition of injections is an injection. I want to know if this is a good proof: Composition of injections is an injection. Let $f:S_1\rightarrow S_2$ and ...
1
vote
3answers
43 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} ...
1
vote
1answer
69 views

It $f(x)=x+\sin x$, then can we find $f^{-1} (x)$?

We have a bijective function $f(x)=x+\sin x$. So what is $f^{-1} (x)$? Let $f^{-1}(x)$ be $g(x)$. Suppose we have to find $g\left(\dfrac{\pi}{6}+\dfrac{1}{2}\right)$ and ...
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3answers
30 views

Nth value of Function

Given x and y we define a function as follow : f(1)=x f(2)=y f(i)=f(i-1) + f(i+1) for i>2 Now given x and y, how to calculate f(n) Example : If x=2 and y=3 ...
6
votes
3answers
118 views

Functions with different codomain the same according to my book?

My book gives the following definition: A function $f$ from $A$ to $B$ is defined as $f\subseteq A\times B$ such that if $(a,b)\in f$ and $(a,b_1)\in f$ then $b=b_1$ and there exists a $(a,b)\in ...
1
vote
1answer
21 views

The second derivative of g(t)= f(x(t), y(t))

For the chain rule for differentiating a function $g(t) = (f(x), y(t))$ how do you get from the identity $g'(t) = \frac{df}{dx} \cdot \frac{dx}{dt}+ \frac{df}{dy}\cdot\frac{dy}{dt}$ to the identitiy ...
1
vote
2answers
49 views

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$.

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$. There is no example like this in my math book.
1
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1answer
59 views

How to prove that $f:\mathbb{N}\rightarrow X$ where $f$ maps to an element in a set, is a bijection?

Let $X$ and $Y$ be disjoint finite sets, $|X|=n$ and $|Y|=m$, so that we have the following bijections: $f:\mathbb{N}_n \rightarrow X$ and $g:\mathbb{N}_m \rightarrow Y$ I need to prove that ...
0
votes
1answer
24 views

How to find the smallest value by using Lagrange multiplicators?

Let $a$, $b$ and $c$ be positive constants. How one can find the smallest value of the sum of three numbers $x_1$, $x_2$ and $x_3$ at the surface $\dfrac{a}{x_1}+\frac{b}{x_2}+\frac{c}{x_3}=1$ by ...
1
vote
2answers
33 views

How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
3
votes
0answers
36 views

Find all differentialbles function [on hold]

Find all differentialbles function $f:[0,\infty)\rightarrow\mathbb R$ such that: a) $f^{\prime}$ is non-decreasing; b) $x^{2}f^{\prime}(x)=f^{2}(f(x)),~\forall x\in\lbrack0,\infty)$
3
votes
2answers
42 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
0
votes
2answers
20 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
0
votes
0answers
20 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...