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

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Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
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42 views

Surface area of a bottle with integration

Would it be possible to model a bottle using a function, then revolving it to determine the surface area and the volume while customizing the curvature and the dimensions of particular sections of the ...
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15 views

show that Lb(a, y) := max{1−ay−by,0}, (a, y) ∈ R×{−1,1}, b ∈ R, is continuouse whit respect to first variable

Show that: $$L_b(a, y) = \max\{1−ay−by,0\},\;\; (a, y) \in\mathbb{R}×\{−1,1\}, b\in\mathbb{R}$$ is continuous whit respect to first variable.
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1answer
15 views

Find the first derivative of given limit

Let $f(x)$ is a polynomial satisfying $f(x).f(y)=f(x) + f(y) +f(xy) -2 $ for all x ,y and $f(2)=1025$ , then the value of lim x tending to 2 $f'(x)$ is I want to know that value at $f(1)=1$ can ...
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2answers
70 views

Difference between $f(x)=\sqrt{x}$ and $f(x^2)=x$

Is there a difference between $f(x)=\sqrt{x}$ and $f(x^2)=x$ for $ x \in\mathbb R^+_ 0 $ ?
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1answer
20 views

2 ways of solving derivative of composition of functions?

Functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $c\left(t\right)=\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} $ I calculate composition and drivative on 2 ways: 1. substitution and 2. chain rule. ...
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1answer
57 views

find all fucntion such $0<|f(x)-f(y)|<2|x-y|$

Find all function $f:N^{+}\to N^{+}$,and for any postive integer $x\neq y$, such $$0<|f(x)-f(y)|<2|x-y|$$ I think $f(x)=cx,c<2?$ it's right?
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2answers
18 views

How do I determine Taylor polynomial of degree 2 of function $g(x,y) = (x^2 + y)e^{xy}$ in development at point $(x_0,y_0)=(1,-2)$?

How do I determine Taylor polynomial of degree 2 of function $g(x,y) = (x^2 + y)e^{xy}$ in development at point $(x_0,y_0)=(1,-2)$? How to do this on easiest way?
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2answers
96 views

Solutions to $[x^2]+2[x]=3x\text{ where } 0\le x\le 2$

Find all solutions to $$[x^2]+2[x]=3x\text{ where } 0\le x\le 2$$ and $[x]=\lfloor x\rfloor$ $$$$ I managed to simplify this to $[x^2]-[x]=3\{x\}$. Thus, $$[x^2]-[x]=\{0,1,2\}$$ However I got ...
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2answers
56 views

Is this a surjection? (Elementary real analysis)

My question is at the bottom of this wall of text, but this text is crucial to the question. I am reading Real Mathematical Analysis by Pugh, and on page 33 he states and proves the following theorem. ...
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2answers
67 views

Prove that $\sin x=[1+\sin x]+[1-\cos x]$ has no solution in $x\in \Bbb R$

Prove that $$\sin x=[1+\sin x]+[1-\cos x]$$ has no solution for $x\in \Bbb R$ where $[x]=\lfloor x\rfloor$ $$$$I reduced the equation into $$\sin x=2+[\sin x]+[-\cos x]$$ From here, I plotted ...
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3answers
42 views

How to transform a straight line into a curve. Linear to convex/concave function.

Let's suppose I have a basic linear function $$f(x)=mx+b$$ I want to factor in a parameter $A$ in a function $g(x)$ such that when $A<0$ in $f*g(x)$, $f(x)$ becomes increasingly concave as $A$ ...
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3answers
55 views

Solutions to $\{x^3\}+\lfloor x^4\rfloor=1$

Find all solutions of $$\{x^3\}+[x^4]=1$$ where $[x]=\lfloor x\rfloor$ $$$$ I know that $0\le\{x^3\}<1\Rightarrow 0<[x^4]\le 1$. Thus $[x^4]=1$. I couldn't get any further though since I'm ...
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6answers
204 views

Prove $\lfloor\frac{n+1}{2}\rfloor+\lfloor\frac{n+2}{4}\rfloor+\lfloor\frac{n+4}{8}\rfloor+\lfloor\frac{n+8}{16}\rfloor+ \dots=n$

Prove $$\left[\dfrac{n+1}{2}\right]+\left[\dfrac{n+2}{4}\right]+\left[\dfrac{n+4}{8}\right]+\left[\dfrac{n+8}{16}\right] + \dots=n$$ where $[x]=\lfloor x\rfloor$ $$$$ It was suggested that ...
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1answer
24 views

Pulse wave formula

I am developing a Game Boy emulator and I need to get a formula for generating pulse waves, like this: (picture from this Wikipedia page) I know that it is possible to generate a square wave with ...
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2answers
67 views

Solutions to $\lfloor 2x\rfloor-\lfloor x+1\rfloor=2x$

Find all solutions to $$[2x]-[x+1]=2x$$ where $[x]=\lfloor x\rfloor$ $$$$ I divided this into 2 cases: $$Case 1:x=[x]+\{x\}\text{ where } 0\le\{x\}<0.5$$ $$Case 1:x=[x]+\{x\}\text{ where } 0.5\...
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1answer
19 views

Composition of function

I have these two functions and I have to do Jacobi matrix of their composition $h\circ c$. $h\left(r,\phi\right)=\begin{pmatrix} rcos\phi \\ rsin\phi \\ r \end{pmatrix} $ $c\left(t\right)=\begin{...
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1answer
45 views

State the range of each function $f:\mathbb{R} \rightarrow \mathbb{R}$

$f(x) = 4\cos(6x)$ $f(x) = -2(x-3)^2+5$ For the first one I just graphed it and found $y = -4$ and $y = 4$ For the second one I also graphed it and found $y = 5$ Did I state the range ...
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1answer
28 views

How to show a continuous function from a space to a subspace is continuous from a space to the whole space?

Let $(X,\mathcal{T})$ and $(Y, \mathcal{J})$ be topological spaces. Let $W \subset Y$ be a subspace with its subspace topology. Show that if $f: X \to W$ is a continuous function, then $f: X \to ...
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0answers
57 views

Spivak problem 26-4 (Calculus 3rd edition)

I am having difficulty understanding a problem requirements from Spivak's for Chapter : "Complex Functions". The problem descriptions is as follows: In this problem we will consider only ...
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2answers
63 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
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0answers
46 views

Simplify $f(x)=\Gamma(n/2)/(\Gamma(1) \Gamma(n/2-1))$… a Rational Expression using the Gamma Function.

I was reviewing a document about an algorithm wherein it is stated that $f(x)$ is a probability density function: (1)$$ f(x)=\frac{\Gamma(\frac{n}{2})}{\Gamma(1)\Gamma(\frac{n}{2}-1)}\frac{2}{n-2}\...
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2answers
127 views

Solutions to $\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$

Find all solutions to $$\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$$ $$$$ Unfortunately I have no idea as to how to go about this. On rearranging, I got $$3\lfloor ...
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1answer
47 views

prove that a function is monotonically increasing

I want to show that the function $ f(b)= \dfrac{2^a-2^b}{\ln(a)-\ln(b)}$ is an increasing function of $b \in [0,a)$, where $a$ is a real constant. I have tried the following things: To prove that ...
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1answer
18 views

Graphing function according to its conditions/limitations

A continuous function is defined by the following conditions $\frac{d^2y}{dx^2}>0 $ for $ x<-1$ and $1<x<3$ $\frac{dy}{dx}=0 $ only when $ x=-3,1$ and $ 5$ $y = 0$ only when $x=1$ ...
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1answer
12 views

Continuous function to describe a shifted Gaussian curve

We have a set of numerical data that strictly follows Gaussian function (e.g. Fig. 1). Suppose if we shift the left half of the Gaussian curve to to its right end (see Fig. 2), trend lines based on ...
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2answers
87 views

What is the difference between $(f^{-1})^{-1}(A)$ and $f(f^{-1}(f(A)))$?

I asked a question Under what condition does $f(f^{-1}(f(A))) = f(A)$? and it totally backfired because people were confused whether $f^{-1}$ is the preimage or the inverse function Let $f: X \to Y$ ...
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3answers
103 views

Under what condition does $f(f^{-1}(f(A))) = f(A)$?

Basic question regarding function. Let $f: X \to Y$, then for what $f$ does $f(f^{-1}(f(A))) = f(A)$? hold? Obviously this relationship holds when $f$ is a bijection. This does not hold when $f$ ...
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2answers
79 views

Solution of $x-1=(x-\lfloor x \rfloor)(x-\{x\})$

Find all solutions for $$x-1=(x-\lfloor x \rfloor)(x-\{x\})$$ $$$$My approach: $$x-1=\lfloor x \rfloor\{x\}$$ $$\dfrac{x-1}{\lfloor x \rfloor}=\{x\}$$ $$\Rightarrow 0\le \dfrac{x-1}{\lfloor x \...
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2answers
95 views

The graph of the function $f(x)= \left\{ \frac{1}{2 x} \right\}- \frac{1}{2}\left\{ \frac{1}{x} \right\} $ for $0<x<1$

Let for reals $$\{x\}=\text{Frac}(x)$$ the fractional part function, take for example the more common definition, the first (there is a different definition as you see in this MathWorld's Page, ...
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1answer
33 views

Find out the function based on a specific rule (Non-linear function)

If I had the table of $x$ , $f(x)$ pairs, and this is the rule: \begin{array} {|r|r|} \hline 1 &1 \\ \hline \hline x_i & f(x_i) \\ \hline \hline 2x_i & 1 +f(x_i) \\ \hline \end{array} ...
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3answers
80 views

Solving the Inequality $\dfrac{x-1}{\lfloor x \rfloor}\ge 0$

Find all solutions of $$\dfrac{x-1}{\lfloor x \rfloor}\ge 0$$ $$$$ I know how to solve the Inequality $\dfrac{x-1}{ x }\ge 0$ using the Wavy-Curve/Method of Intervals technique. However I don't know ...
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2answers
48 views

Find the integral values of $a$ for which $f(x)$ is onto.

A function $f:\mathbb R\rightarrow\mathbb R$ is defined by $f(x)=\dfrac{ax^2+6x-8}{a+6x-8x^2}$. Find the integral values of $a$ for which $f$ is onto(surjective). My attempt: As given in question ...
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0answers
33 views

Notation of a function that Maps two sets into a Matrix

Given two sets $P, V$ a function $f(t)$ takes any element that belongs to $ P $ or $ V $ e.g. $ t \in P \cup V$ returns a matrix of $ 2 $ columns and $K$ rows. What is the proper notation to express ...
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2answers
49 views

$f$ in $f(x)$ as a vector

I might split this question into two, so the first paragraph will contain the main question. Given a linear function $f(x,y)$, is it possible to consider $f$ as a vector? Given the relationship of ...
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1answer
43 views

How to find a basis for subspace of functions

I am doing this exercise: The cosine space $F_3$ contains all combinations $y(x) = A \cos x + B \cos 2x + C \cos 3x$. Find a basis for the subspace that has $y(0) = 0$. I am unsure on how to ...
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1answer
40 views

Converting parametric function into cartesian

I am trying to convert the parametric function $x(t) = a\cdot(t - \sin(t)) + b\cdot\cos\left(\frac{t}{2}\right)$ $y(t) = a\cdot\cos(t) + b\cdot\sin\left(\frac{t}{2}\right)$ into a cartesian form. ...
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3answers
104 views

Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
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1answer
39 views

Is this $2d$ function injective and/or surjective?

Consider a function $f:\mathbb{R}^2 \to \mathbb{R}^2$ defined by $f(x,y)=(x,xy)$. Is this function injective, surjective? I can figure out that this is injective but cannot prove it surjective. ...
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1answer
36 views

Piecewise Functions difficulty

I am new here. I need a bit of help. Not the best at math. Johnny jumps from a plane at $3000$ feet in the sky. After $35$ seconds, he deploys a parachute. $h =$ height. $t =$ # of seconds after ...
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3answers
798 views

If $f:X\rightarrow Y$ and $V\subseteq Y$, does there exist $U\subseteq X$ such that $f(U)=V$?

I am afraid this is quite a trivial question. However, let $f:X\rightarrow Y$, where $X$ and $Y$ are sets. Is that true that for any subset $V$ in $Y$ there exists a subset $U$ in $X$ such that $f(U)=...
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19 views

Increasing induced functions.

I am studying partial ordered sets. I have a problem with the following example: $ \text{Let X and Y be sets and } f \in Y^X. \text{The induced functions } f:P(X) \to P(Y)$ and $f^{-1}:P(Y)\to P(X) \...
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1answer
41 views

Unconventional Differentiation Rules

We all know the stock-standard and conventional differentiation rules, such as the Sum and Difference Rule, Product Rule, Chain Rule etc. But are there other more advanced rules that are not treated ...
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1answer
32 views

Solution of equation

If $f(x) = x^2 - 2ax + a(a+1)$ , $f:[ a, \infty] \to [a,\infty]$ . If one of the solution of the equation $f(x)=f^{-1}(x)$ is $5049$ , then what may be the other solution ? My WORK: I found the ...
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22 views

Find the number of into functions A-A={1,2,3,4,5) such that f(i)≠i [closed]

It can be tried by using total no. of functions such that f(i)≠i-total no. of onto functions. total no. of such functions will be= 4^5 total no. of onto functions= D5( dearrangement)
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2answers
39 views

Range when two piecewise functions are composed

$$f(x):=\begin{cases} [x] & -2≤ x ≤ -1 \\ |x| +1 & -1≤ x ≤ 2 \end{cases}$$ $$ g(x):=\begin{cases} [x] & -π ≤ x ≤ 0 \\ \sin(x) & \...
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2answers
38 views

Minimum number of possible value of x

Let $f: R\rightarrow R$ is a function stratifying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)$ for all $x$ belonging to $R$ . If $f(0) =5$ , them minimum number of possible value of $x$ satisfying $f(x) =5$ for ...
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1answer
51 views

Is there any equation for this type of skewed parabola?

I am having the following parabola looking curve (blue curve), but its not exactly symmetric. The best fit that I am getting using a quadratic equation is also shown (black curve) but it is not ...
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0answers
29 views

Is there a complete function for the integers?

Completeness is used to describe many things, but in logic it is often used to describe a set of operators from which any input-output mapping can be created. Does there exist a function which can ...
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25 views

Question about composed functions [closed]

Let X be a finite set and $f : X → X$ a function. Let $f^1 = f$ and if $f^ n$ has been defined for n ∈ N then set $f^{n+1} = f ◦ f^ n$ . a) Prove for some n, $f^{n+1} = f^{n}$. b)Set Y = $Range(f^n)$ ...