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

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0
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0answers
9 views

How to create a function for residual sum of squares in matlab

I need to write a matlab loss function for the residual sum of squares function. in particular: $$G(\beta)=\sum_{i=0}^n (y_i-X\beta)^2 $$ where y is a 60 x 1 vector x= matrix of 5 vectors ...
0
votes
2answers
17 views

Bound on oscillation of product of functions.

For $f,g:\mathbb{R}^n\to [-M,M]$, prove that $$\rm{Osc}_{fg}\leq M(\rm{Osc}_f+\rm{Osc}_g)$$ Where $$Osc_{f}(x_0)=\lim_{\varepsilon\to0}(\sup_{x\in B_\varepsilon(x_0)}f(x)-\inf_{x\in ...
0
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1answer
22 views

Definition of $n$-to-$1$ mapping.

What is the definition of "$n$-to-$1$ mapping"? Does an $n$-to-$1$ mapping mean to say that if $f$ is a function from $A$ to $B$, then for every $y\in R(f)$ there exists $n$ different elements in $A$ ...
6
votes
1answer
54 views

Range of function $f(x) = \sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$

Range of function $f(x) = \sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$ $\bf{My\; Try::}$ For $\min$ of $f(x)$ $$\left(\sqrt{13-x}+\sqrt{x}\right)^2=13-x+x+2\sqrt{x}\sqrt{13-x}= 13+2\sqrt{x}\sqrt{13-x}\geq ...
0
votes
3answers
31 views

How to find the range of this specific function? [on hold]

$$f(x) = (x-1)e^{x+2} - \frac {x^2}2 $$ if x is in the domain of $[-2,1]$ Image of the function Thank you.
0
votes
2answers
37 views

If $f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$

If $\displaystyle f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$ $\bf{My\; Try::}$ Here Domain of $\arcsin x\;,\arccos x$ is ...
11
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1answer
87 views

How to find the inverse of $f(x)=x+ \frac{x^{3}}{1+x^{2}}$?

I know that given, $f(x)=x+ \frac{x^{3}}{1+x^{2}}$ I should set $y=x+ \frac{x^{3}}{1+x^{2}}$ and solve in terms of $x$, then just swap the $x$'s and $y$'s. I know that, since the derivative is ...
0
votes
1answer
13 views

Impose $f$ is function of $x$ in MAPLE

I am a bit new in MAPLE and I was trying to write a function of $x$. I define a function: $W:=W_{d_{{\it ij}}}\,W_{f_{{\it ij}}}$ I would just like to write in MAPLE $diff(W,x)$ and get: $ ...
0
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1answer
23 views

Finding a function with a certain behavior

I'm searching what are the keywords or the good links to expand my researches. I would like to get the equation ( for programming purposes ) of more or less this curve: ...
2
votes
0answers
32 views

Modular Algebra

I am devising an algorithm to solve equations like the following: $$10^{\lfloor\log(p1)\rfloor}x+p_1\equiv0\pmod{p_2}$$ In the scenario: $10^{1}x+5\equiv0\pmod{7}$, where $p_1=5$ and $p_2=7$, ...
0
votes
0answers
15 views

What is Burl’s monthly payment, and how many payments will he make? [on hold]

Burl uses the TVM Solver to estimate the monthly payment for his mortgage. The TVM Solver screen shows these settings: N=180, I%=5.3, PV=250 000.00, PMT=-2008.70, FV=0.00, P/Y=12.00, C/Y=2.00
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votes
1answer
62 views

how can I find the convergence of the integral $\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx$ , for $ x \in [0,1]$ [on hold]

I want to check the convergence of the integral $$\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2} dx $$ for $ x \in [0,1]$ and n->∞ is a constant so can basically pulled out of the integral but I don't know ...
2
votes
2answers
55 views

A question about alternate series involving unit fractions

I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about. Let $A\subseteq\mathbb N$ be a subset that contains at least $n$ elements; ...
0
votes
1answer
15 views

Two variable function - Convex/concave

Consider the function: $f(x,y)=e^{ax+by^{2}}$ I have to find the values for $b$ such that $f(x,y)$ is convex and concave. These are my calculations: $f_{xx}=a^{2}e^{ax+by^{2}}$ ...
2
votes
0answers
36 views

How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - ...
1
vote
1answer
30 views

Find the range of the function $f(x) = 4x + 8$ for the given domain $D = \{-5, -1, 0, 6, 10\}$

The question is to find the range of each function for the given domain $f(x)=4x+8$, $D=\{-5, -1, 0,6, 10\}$. Is the range just $R= \{-12,4,8,32,48\}$ or am I mistaken? Could you elaborate why my ...
0
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0answers
15 views

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve ...
0
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0answers
21 views

$\log(x)$ as iteration-series: how can this be made correct?

I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question. One idea ...
2
votes
2answers
76 views

Find an injective function that maps $\mathbb{R} \to (-\infty, 0]$

I'm looking for any ideas as to a function which maps $\mathbb{R} \to (-\infty, 0]$. I considered $-|x|$ but realised that is not injective.
1
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3answers
64 views

how can I find the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , $ x \in (-1,1)$ [on hold]

I want to check the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , for $ x \in (-1,1)$ but i don't know what to do. Every theory I know it is not working. Can someone ...
4
votes
1answer
117 views

A continuous onto function from $[0,1)$ to $(-1,1)$

How I can construct a continuous onto function from $[0,1)$ to $(-1,1)$ ? I know that such a function exists and also I have a function $\displaystyle f(x)=x^2\sin\frac{1}{1-x}$ which is ...
0
votes
1answer
45 views

Clarification of the topology lemma “Any continuous and open injection of the open disk extends over the circle”

My elementary topology 1 class last semester used the book "Topology: Point-Set and Geometric" by Paul Schick, and covered through the end of chapter 8. I am working through the rest of the book on ...
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0answers
23 views

what is Step function?A general definition of what it is. .And explain about it please. [on hold]

what is Step function?A general definition of what it is. .And explain about it please.
1
vote
3answers
135 views

Derivative of the following function (similar to Softmax)

I am having a hell of time trying to differentiate the following function with respect to x. Do you have any suggestions $f(x) = \frac{ w(i)^x}{ \sum\limits_{j} w(j)^x }$ where $w$ is a vector ...
0
votes
1answer
22 views

Derivatives relation

Please, help me solve the following problem: Suppose $$\frac{df(x,y)}{dx}>0, \qquad \frac{df(x,y)}{dy}<0, \qquad \frac{d^2f(x,y)}{dxdy}<0$$ Is it true that if $y_{2}>y_{1}>0$ then ...
2
votes
3answers
17k views

How many one to one and onto functions are there between two finite sets?

Suppose $X$ has $N$ elements and $Y$ has $M$ elements. How many one to one function are there from $X$ to $Y$? How many onto function are there from $X$ to $Y$? The number of one to one functions ...
0
votes
1answer
10 views

What if the input of a simple function question is X?

I know how to answer function questions when they are like: fg(3) when f(x) = x + 3 and g(x) = x^2 But what do I do when the question is like: fg(x) when f(x) = x + 3 and g(x) = x^2 Or for a ...
0
votes
1answer
423 views

How to solve for maximum area of a rectangle under a curve?

Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is $8^{-\frac{x}{5}}$. I would greatly appreciate a full explanation. I already ...
4
votes
4answers
71 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ ...
0
votes
1answer
22 views

Determine a curves position over another curve

if the curve of $y= mx^2 -2mx +m$ is over rhe curve of $y=2x^2 -3$, then the limits of the interval must be my attempt: I dont know which concept i have to use. I only know that discriminant is use ...
0
votes
1answer
14 views

Finding the upper tight bound of a mathematical function. (Big O)

I am trying to understand Big-$O$ notation through a book I have and it is covering Big-$O$ by using functions although I am a bit confused. The book says that $O(g(n))$ where $ g(n)$ is the upper ...
3
votes
4answers
75 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
1
vote
2answers
29 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
5
votes
2answers
400 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
1
vote
1answer
56 views

Engineering/mathmatics question

I have an equation $M(x)= -15.328x^2+176.44x-352.88$ (a parabola) and also $V(x) = -30.657x + 176.44$. I want to know how to find $x$ where the values of $M$ and $V$ combined are the lowest, I'm ...
1
vote
0answers
23 views

Point which do not lie in domain of $f (\frac{2}{x-2})$

If $$f(x)=\frac{1}{x^2-17x+66}$$ then the points which are not in the domain of $f (\frac{2}{x-2})$ are: $(A) \frac{7}{3}$ $(B) \frac{24}{11}$ $(C) \frac{8}{3}$ $(D) 2$ I have already found that ...
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1answer
23 views

Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$

Let $f:X \to Y$ be a map with $A_1,A_2 \subset X$ and $B_1,B_2 \subset Y$. Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$ where $f^{-1}(B) = \{x \in X: f(x) \in B\}$. Attempt: ...
0
votes
2answers
116 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...
2
votes
2answers
77 views

How many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. A few related questions inside.

I am trying to calculate how many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. I would like to see the explicit mappings and learn how you ...
-1
votes
0answers
24 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
1
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0answers
47 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
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3answers
37 views

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$. [on hold]

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$.
0
votes
0answers
8 views

Find the maximum value of $f(x,y,z)$ on the interval $x_0<x<g^x(p)$, $y_0<y<g^y(p)$, $0<z<g^z(p)$, $p=p(x,y,z)$

First of all, sorry if I am misusing terms or any tags in the post; I am a bit out of my depths here so I'm just trying to explain things in layman's terms. Now, here's the problem: I am working on ...
0
votes
0answers
10 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
0
votes
0answers
29 views

How to prove a function is not positive definite [on hold]

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
1
vote
2answers
32 views

Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto

Let $f:A \to B$ and $g:B \to C$ be maps. Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto. Attempt: If $g \circ f$ is onto, then for all $y \in A$, $\exists x$ such ...
0
votes
1answer
33 views

Hi I was wondering if there is any algebraic way to find the zeroes of a cos/sin formula without using the unit circle? [on hold]

I understand how to find the zeroes using the unit circle or just graphing it for any matter for my equation. I was just wondering if there is a formula or an algebraic way I could find them. ...
4
votes
1answer
18 views

Constructing bijection from set of equivalence classes to another set

Suppose $f:A \to B$ is surjective. Define a relation on $A$ by setting $x\sim y$ if $f(x) = f(y)$. It is clear that $\sim$ is an equivalence relation on $A$. Let $\mathcal{E}$ be the set of ...
1
vote
1answer
158 views

How can functions disagree with the values of its expansions at some points on an algebraic curve

I found a curve, in which some function has at least two expressions, which differ infinitely much!! Is there any error in the thoughts? The curve is defined by "\begin{equation} ...
1
vote
3answers
221 views

Range of a complicated function

Is there any way to figure out the range of values of the function $$y=\frac{2}{x}\cdot \sin(x)?$$ The domain is so easy to know. It's all real numbers except $0$. However the challenging part is to ...