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

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-1
votes
2answers
256 views

Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$

hello everyone I have question Q Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$? Notice, (the inverse image not inverse ...
0
votes
3answers
267 views

How to derive a multi-dimensional function?

In a regular function it possible to derive it by - $$f'(x_0)=\frac{f(x_0+h)-f(x_0-h)}{2h}$$ when $h$ is "little enough" . How could I derive a multi-dimensional function with the above approch ? ...
3
votes
3answers
94 views

Given a functor between categories, how to denote a morphism between particular objects of that category

I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that: there is a functor between categories, $f:C_1\to C_2$, $c_1$ is ...
0
votes
2answers
88 views

How to solve functions of type $f(g(t))$

Please can someone guide me how to solve this type of problem? If $f$ is the function $f(x) = 1 +x +x^2$, find $f(g(t))$. I've never solved them before. I just need an idea.
2
votes
3answers
95 views

Solve for $x$: $4x = 6~(\mod 5)$

Solve for $x$: $4x = 6(mod~5)$ Here is my solution: From the definition of modulus, we can write the above as $ \large\frac{4x-6}{5} = \small k$, where $k$ is the remainder resulting from ...
3
votes
2answers
3k views

Prove that the only eigenvalue of a nilpotent operator is 0?

I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How ...
2
votes
1answer
33 views

Condition for differential inequality

Let $f(x) = \frac{e^{ - ax}}{1 + {e^{bx}}}$, where $x>0$, $a$ and $b$ are positive constants. Find the condition of $a$ and $b$ so that $$ ( - 1)^nf^{(n)}(x) \ge 0 $$ with all $x>0$ and $n$, ...
1
vote
3answers
95 views

equivalent function for this graph

$ \forall x\in (0,n] $ $ f(x) = \begin{cases} 1, & \text{if $x$ $\in$ (0, 1]} \\ x, & \text{if $x$ $\in$ (1, 3]} \\ 3, & \text{if $x$ $\in$ (3, n]} \\ \end{cases} $ How can I ...
5
votes
2answers
354 views

$f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$

How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$ ($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
1
vote
2answers
580 views

Easing function, constant velocity then decelerate to zero

I'm trying to write an interpolator for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an ...
2
votes
5answers
2k views

How many types of functions are there [closed]

We have the following types of functions : a) Logarithmic function b) Rational Function c) Irrational Function d) Piecwise or modulus function e) Smallest integer function or cieling function f) ...
1
vote
1answer
26 views

Function for a series of values

What is the best way to find a function based on a series of data points? The data I have so far: fx(1,000) = 30,000 fx(10,000) = 8,000 fx(100,000) = 1,500 fx(1,000,000) = 300 fx(10,000,000) ...
3
votes
2answers
56 views

Functional relations : Trouble seeing transitivity

Given the following domain: $\;\{1,2,3,4\}$ And the following relation: $$\{(1,1),(1,3),(1,5),(2,2),(2,4),(3,1), (3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)\}$$ It states that this is an equivalence ...
1
vote
1answer
54 views

How do we determine as to how long we should sum an asymptotic series of a function to get the answer correct up to a particular precision?

As an example, consider the asymptotic expansion for polygamma function . What should be the min value of 'k' in the equation to get the answer correct upto a particular precision, say pth. Is there ...
2
votes
1answer
41 views

Functions: Detirmining values a & b

The problem $f(x)$ and $g(x)$ are defined over the real number set $\mathbb{R}$ as follows: $$ \begin{split} g(x) &= 1-x+x^2\\ f(x) &= ax+b \end{split} $$ If $g(f(x)) = 9x^2 - 9x + 3$, ...
5
votes
2answers
108 views

How to exactly write down a proof formally (or how to bring the things I know together)?

I've always had trouble with this: I know everything I need to know but I can't connect everything I know in the way it's being wanted from me. This is what I have to do: Prove for $f : M → N$ the ...
5
votes
2answers
128 views

If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.

Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that $g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
1
vote
2answers
104 views

find maximum and minimum for any function

I'm writing an optimization algorithm thats supposed to find the maximum and minimum value of any given function. Whats the fastest numerical approuch to do so?
2
votes
3answers
226 views

Differential calculus - Reviewing and drawing graph

I have missed math class for a few weeks and I'm quite behind with the new stuff learned by the others, so I'm stuck with a problem here. The main problem is, I'm going to have hard time explaining ...
0
votes
0answers
30 views

Velocity and Distance; Functions of Time

$s(t)$ = distance a particle travels from time $0$ to $t$. If in this case, the distance $s$ is only the function of time then its necessary that the velocity should be constant. Likewise, $v(t)$ = ...
0
votes
2answers
47 views

Distance of a particle; a function its time

Force is a function of mass and acceleration. Here mass is a fundamental quantity, and acceleration is a derived quantity OR $F(a, m) = ma$. I want to ask that why the distance traveled by a ...
1
vote
1answer
56 views

How can I prove this function equation

let $g:[a,b]\longrightarrow [0,1]$ and $g$ is one to one and is increasing,prove that there exists a funcition $f$,such that $$g(x)=\dfrac{f(x)-f(a)}{f(b)-f(a)},x\in (a,b)$$,where $f$ is one to one. ...
1
vote
2answers
496 views

Values for which functions are undefined

If a function is given by the rule $f(x) = x +1$ and we declare any three real numbers lets say, $2$, $3$, and $4$ as the permitted inputs or the domain of the function, then $$f = \{(2, 3), (3, 4), ...
3
votes
1answer
78 views

Showing a bijection with a contraction

I have the function $F(x) = x + f(x)$ where $f(x)$ is a contraction: $|f(x)-f(y)| \leq \alpha|x-y|$ for some $0 < \alpha < 1$ and all $x, y \in \mathbb{R}$ I want to show that $F$ is a ...
0
votes
0answers
37 views

Help with functions, confirming if I'm correct.

Let $\mathscr F$ denote the set of all functions from {1, 2, 3, 4} to {1, 2, 3, ... , 10}. a) Find and simplify the number of functions $f \in \mathscr F$ so that f(1)=1 and f(2)=2. b) Find and ...
0
votes
1answer
54 views

vectors and arc length

I have a short questions. I have two functions that describe an arc. One describes the vertical movement, one the horizontal movement. If we say that $$ y(x)= C_1\cos{\frac{\pi x}{2L}} \\ z(x)= ...
0
votes
2answers
55 views

Where is this function welldefined?

Can this function $$\left(\frac{3a^2-a}{15+3a}\right)\left(\frac{25-a^2}{9a^2-6a+1}\right)\left(\frac{3}{a}-9\right)$$ be simplified to $$\frac{(5-a)(1-3a)}{(3a+1)}$$ and if that is true a can't be ...
0
votes
1answer
51 views

Help proving and counting functions.

Let $\mathscr F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$. a) Of the two following statements, one is true and one is false. Prove the true statement. Write out the ...
1
vote
2answers
104 views

non continuous ivt problem

solving for an ivt for a non-continuous function We take $f : [0,1] \to [0,1]$ a non-increasing function, such that $f(x)≥ f(y)$ whenever $x≤ y$; and we want to prove that there exists $c\in [0,1]$ ...
1
vote
4answers
126 views

Injection / Surjection

Here's the question I got that it is injective. By saying: $f(y) = 3 - y^2$ Suppose; $f(x) = f(y)$ $3 - x^2 = 3 - y^2$ $x^2 = y^2$ $sqrt(x) = sqrt(y)$ $±x = ±y $ I conclude the it's not ...
0
votes
1answer
62 views

Finding a function which fits this data?

I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
1
vote
2answers
65 views

correct name of mathematical property

I am developing a program that transforms artifacts in one (computer) language to artifacts in another language. In my program there are certain border line situations where the result of applyin the ...
0
votes
2answers
71 views

one-, to-one, and onto functions

What determines a function as one-to-one, and onto? And what would this function be classified as? $A = B = \Bbb Z, f:A\to B$ $f(a) = a-1$ Little help please?
0
votes
1answer
28 views

The intersections of two equations

I have these two functions, that I must solve. When I plot them, I see there are 4 intersections: $$(1,0),(0,1),(-1,0),(0,-1)$$ But how do you solve these??
1
vote
1answer
32 views

Visualising a graph from $\mathbb{R^3}$ to $\mathbb{R}$

Given a function, any function, exempli gratia, $$f(x,y,z)=\frac{x^2y}{z}$$ It's range will be whole of $\mathbb{R}$(???). Plotting the graph does not make sense. How can I, if I can, visualise this ...
2
votes
1answer
20 views

Proof Regarding Series of Functions.

If it's not too much trouble, may I have some help on this question regarding series of functions? Let $u$ and $v$ be two series of functions on a set $X$ such that $|u| < |v|$ for every $x$ ...
4
votes
4answers
324 views

Why $f^{-1}(f(A)) \not= A$ [duplicate]

Let $A$ be a subset of the domain of a function $f$. Why $f^{-1}(f(A)) \not= A$. I was not able to find a function $f$ which satisfies the above equation. Can you give an example or hint. I was asking ...
1
vote
0answers
55 views

Probability function (Very simple indoor positioning system)

I am a programmer, I am working on a masters thesis, tts an Android application, where I will need to implement a very simple indoor positioning system using WiFi RSS. The number of routers is fixed. ...
1
vote
4answers
220 views

Is there a bijective function where $f: \mathbb Q \to \mathbb Q \setminus \{0\}$

$f(x) = {1\over x}$ should be wrong, as the function isn't defined for $0$. Another could be: $f(x) = 2^x$, but is there anything else except functions of this type?I was thinking of something with ...
0
votes
1answer
34 views

Changing intervals of a function

When the intervals changed in the function rule to $(-\pi,0)$ why did $2\pi-x$ become $-x$? $$f(x)=\begin{cases} x & 0\le x \le \pi\\ 2\pi-x & \pi < x < 2\pi \end{cases}$$ ...
1
vote
6answers
143 views

Proof that if $\forall a f(a) = g(a)$ then $f=g$

How do we prove formally that if: $\forall a f(a) = g(a)$ $=>$ $f=g$ when $f,g \in \mathbb F[x]$
0
votes
2answers
78 views

Proving bijectivity f: ℚ² → ℚ²

This is what I have to proof bijectivity for: f: ℚ² → ℚ² : (x,y) ↦ (3x + y, x + 2y) First I have to proof that the function is injective by doing: f(x,y) = f(x',y') And that's where I ...
2
votes
1answer
303 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
3
votes
1answer
107 views

Find the values of $x,y$ and $z$ in these equations

I'm stuck with these equations. Can somebody help me with solving it? If $$(a+b) = \frac{x(y+z)}{x+y+z},\quad (b+c) = \frac{y(x+z)}{x+y+z} ,\quad (a+c) = \frac{z(x+y)}{x+y+z}$$ then find $x,y,z$.
1
vote
1answer
243 views

Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality

I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence Let $A$ be ...
3
votes
1answer
124 views

Is this claim true that $g\circ h$ is bijection?

Please help me to probe the truth of the following statement. if $g:Y \to Z$ , $h:X \to Y$ and $g\circ h$ is bijection $\Rightarrow$ $g$ and $h$ are bijection too.
0
votes
1answer
73 views

Linear Algebra Proof of Injective Function

I'm new in the University and I don't know how to solve this: Suppose $v$ is a non null element of a vector space $V$ on $\mathbb R$. Show that the function is injection: $\mathbb R\to V $ $t ...
1
vote
1answer
253 views

Comparing rates of change: which function increases faster?

I am comparing two functions for $x \ge 1$: $$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$ $$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
5
votes
3answers
382 views

Continuous function that take irrationals to rationals and vice-versa. [duplicate]

Can someone help me? How can I prove that there isn't an everywhere continuous function $f:\mathbb R \rightarrow \mathbb R$ that transforms every rational into an irrational and vice-versa?
2
votes
4answers
315 views

Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, … $ but is continuous at every other point

(a) Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, ... $ but is continuous at every other point. (b) Find a function $f:\Bbb R \to \Bbb R$ which is ...