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

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-4
votes
1answer
33 views

Let f be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. [closed]

Let $f$ be the function from $A = \{a, b, c, d\}$ to $B = \{e, f, g, h\}$ given by $f = \{(a,e), (b,f), (c,g), (d,g)\}$. If $D = \{a,b\}$, what is $f(D)$? If $G = \{f,g\}$, what is $f^{-1}(G)$? If ...
1
vote
0answers
35 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
2
votes
1answer
65 views

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$ My attempt: I've attempted to solve this question, but isn't it impossible to solve, i.e has already been simplified completely? ...
22
votes
12answers
8k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
0
votes
1answer
26 views

find an injective function from a finite set to an infinite one, and a surjective inverse

I have to prove that there exists an injective function from $X$ to $Y$, being $X$ a finite set, and $Y$ an infinite set. I must also prove that there exists a surjective function from $Y$ to $X$. My ...
-1
votes
1answer
52 views

Solve $f(-2x+3)=-f(4x-9)$ [closed]

Consider a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective, has $f(\mathbb{R})=\mathbb{R}$ and $f(-1)=0$. Can the equation $f(-2x+3)=-f(4x-9)$ be fully solved? If yes, provide a proof. If not, ...
-3
votes
0answers
16 views

Definition: f : A ! B is one-to-one if 8t 2 B, 9 at most one s 2 A such that f(s) = t. [closed]

The question ask me to prove by definition of one-to-one if this statement is one-to-one? I prefer False since it has nothing to do with one-to-one function. I'm I correct.
0
votes
0answers
15 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
2
votes
1answer
23 views

Expectation of the fraction a random function covers its range

Preamble: The number of onto functions from a set of $m$ elements to a set of $n$ elements is, as stated in this answer, computed as follows: $$n!{m\brace n}\;.$$ Now, let's count the number of ...
3
votes
0answers
22 views

Let $A$ be a set with $m$ elements and let $B$ be a set with $n$ elements where $m,n\in \omega$ and $m>n$. If $f:A\to B$, then $f$ is not injective

So I am still learning how to work with infinite sets, and this particular problem is giving me some issues. Right now, I am trying to pick some $x_1,x_2\in A$ such that $f(x_1) = f(x_2)$ to serve as ...
1
vote
1answer
36 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ ...
-1
votes
0answers
28 views

Function “all part” of $x$? [closed]

There is a simple math task that I need to solve: $D(f)=[0,1]$ Solve the $f([x])$. The problem is that I don't know what exacly means $[x]$ (all part of $x$)?
-1
votes
1answer
14 views

Two variable function fulfilling specific conditions [closed]

Does a two variable function $f(x,y)$ exists which satisfies the following conditions: $$ \lim_{x\to\infty}f(x,y)=1$$ and $$\lim_{x\to y}f(x,y)=1$$
0
votes
1answer
23 views

Showing surjection

Suppose $\psi$ is a bijection $\psi: G\rightarrow H$ $\left ( g \right )\psi \mapsto h$ I want to show that $\psi^{-1}$ is also a bijection. $\psi^{-1}: H\rightarrow G$ $\left ( h \right ...
-4
votes
1answer
127 views

Sorting the digits of $\pi$ [closed]

Given a function that sorts the digits of a real number, $\operatorname{sd}(r) \rightarrow r$, examples: $\operatorname{sd}(1.332) \rightarrow 1.223$ $\operatorname{sd}(32140) \rightarrow 1.234$ ...
0
votes
0answers
19 views

inflexion points of a composition of functions

Let $f,g$ be two, smooth real positive and bounded functions over $\mathbb{R}^{+}$, with $f$ monotonously increasing and $g$ monotonously decreasing and both $f$ and $g$ have a single inflexion point. ...
0
votes
1answer
23 views

I need someone to show me how to solve this input/output problem

Alright, so I have: $4y^3 = x$ And now I have to solve for $y$, where I can later use that equation to answer other questions I have. Can someone hint me out on how to solve for $y$ given the above ...
0
votes
1answer
19 views

Intersection of an exponential function and its inverse [closed]

Given the function $$f(x) = 2e^x - 4$$ I found it's inverse $$g(x) = \ln((x+4)/2).$$ I now need to find the intersection of these two ...
2
votes
5answers
155 views

Why this function is continuous and not differentiable at point $x=1$

I have a function $$f(x) = \begin{cases}x^2+2,& x\leq 1\\x+2 ,& x > 1\end{cases}$$ I have to show that this function is continuous and not differentiable at point $x=1$, but when I look for ...
-2
votes
0answers
10 views

A question on limits and integrals of continuous functions.

Assuming a function is continuous is it possible to do this, $$ \lim_{n\to\infty} \int_0^1 f\left(\frac{x_1+x_2+..+x_n}{n}\right) dx_1dx_2...dx_n = \int_0^1 ...
0
votes
2answers
39 views

Is this function one-to-one or onto?

Let $c:P(\mathbb{N})\to\mathbb{Z}$ be the function defined by $c(X)=−1$ if $X$ is empty and $c(X)=\min_{x\in X} x$ if $X$ is not empty. Is $c(X)$ one to one? Explain. I have asked this question ...
35
votes
4answers
4k views

Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
3
votes
2answers
27 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
0
votes
1answer
32 views

Let f(x) = 2x. What is i. $f(\mathbb{Z})$? ii. $f(\mathbb{N})$? iii. $f(\mathbb{R})$?

I am struggling with this question and really need hints I wish if there is an upload file button here so that I can upload my solution to the question for check, since i faced difficulties typing ...
-1
votes
2answers
26 views

Let $A =\Bbb N − \{n^2 : n \in\Bbb N\}$. Construct a bijection from $A$ to $\Bbb N$.

To solve this function, I was thinking to use an index if that's possible: $f(x) = (i = 1$ if $x = 2$ --------$(i +1$ if $x_n = x_{n-1} + 2$ --------$(xi/x$ I see no other way to keep increasing the ...
1
vote
1answer
27 views

Let $f(x) = ax + b$ and $g(x) = cx + d$, where $a, b, c, d$ are constants. Determine for which constants $a, b, c, d$ it is true that $f ◦ g = g ◦$

I'm working on this question and this what I did I get $f•g(x) = f(cx+d)=a(cx+d) + b = acx +ad + b $ $g•f(x) = g(ax+b) = c(ax+b) + d = acx + cb + d $ So how to I get $f•g = g•f$?
0
votes
2answers
57 views

How to compute $\int \sqrt{x}\sin{\sqrt{x}}dx$?

How to compute $\int \sqrt{x}\sin{\sqrt{x}}dx$? I tried let $u=\sqrt{x}$ and $du=\frac{dx}{2\sqrt{x}}$ and apparently it doesn't work. Some better ideas?
0
votes
3answers
43 views

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$?

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$? So let us set $f(x)=\sin{x}$ then it's differentiable on $(x,y)$ and continuous on $[x,y]$. ...
6
votes
2answers
68 views

How to find $\lim\limits_{x\to 2}f(x)$ if $\lim\limits_{x\to 2}\frac{f(x)-5}{x-2}=100$?

How to find $\lim\limits_{x\to 2}f(x)$ if $\lim\limits_{x\to 2}\frac{f(x)-5}{x-2}=100$? I suppose that we do not use L'Hopital rule here. Then $\lim\limits_{x\to 2}f(x)-5=0$, then $\lim\limits_{x\to ...
0
votes
1answer
65 views

Formal Proof that $f^{-1} \circ f = id_x \ , \forall f$

Given $f$ as an invertible function with domain $X$ and codomain $Y$, then we can say $$f^{-1}(f(x)) = x $$ Or since they are both logically equivalent $$ f(f^{-1}(x)) = x $$ This can also be ...
0
votes
0answers
46 views

Sinusoidal function from $4$ knowns

I am trying to find out what sinusoidal function is represented by a minimum of $(0, -10)$ and a maximum of $(2, -4)$ I know that the period is $4$ and that the midpoint would be $(1, -7)$ but that ...
0
votes
1answer
19 views

Continuity of the function $F(x)=\int_0^{+\infty} f(x,t) d t$

Let $f(x,t): \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ be a smooth function (that is any order of derivative with respect $x$ and $t$ exists). Can we say $$F(x)=\int_0^{+\infty} f(x,t) d t$$ ...
1
vote
0answers
37 views

Accumulating integration problem - help me to understand

Here is the problem stem: A factory produces bicycles at the rate of $ p(w) = 95 + 0.1w^2 - w$ bikes per week for $0 \le w \le 25$. The factory ships bicycles out at a rate of $s(w)= 90 $ for $0\le ...
0
votes
2answers
63 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? Suppose that the linear map $T:U\to V$ is a bijection. So $T$ has an ...
1
vote
0answers
16 views

Find all functions such that $f(m)+f(n)|m^p+n^p$

For fixed prime number $p$, find all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)+f(n)\mid m^p+n^p$ for all $m,n\in \mathbb{Z}^+$ I managed to get only that for prime $q$ we have ...
1
vote
2answers
22 views

How to calculate $G(0),G'(0),G''(0)$ from $G(x)=\int^0_x g(t)dt$?

How to calculate $G(0),G'(0),G''(0)$ from $G(x)=\int^0_x g(t)dt$? I think $G(0)=\int^0_0 g(t)dt$ is just a single point so $G(0)=0$ Also since $G(x)=-\int^x_0 g(t)dt$, so $G'(0)=-g(0)$ and ...
1
vote
0answers
46 views

Need help in deciding the type of function and their range and inverse for the functions $p,q,r$ and $s$.

Let $p,q,r$ and $s$ be the following functions $p: R\to R$ defined by $P(x) = (1/2)x + 1$. $R$ represents set of all real numbers. $q:Z \to \{0,1\}$ defined by $q(x) = \{ 1,$ if $x \geq 1; 0,$ if ...
0
votes
1answer
9 views

Proving a function a locally and globally lipschitz continuous

Consider the function $$f(x) = \begin{bmatrix}x_2\\-x_1+\varepsilon(1-x_1^2)x_2\end{bmatrix}$$ Show that this function is locally lipschitz on the set $D=\{x\in\mathbb{R}^2\,\colon \Vert ...
0
votes
0answers
18 views

Sufficient conditions to have the supremum of a continuous function continuous?

Consider a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ with $\mathcal{X}\subseteq \mathbb{R}^k$ and $\mathcal{Y}\subseteq \mathbb{R}^p$. Under which sets of conditions is ...
0
votes
0answers
13 views

From separate continuity to joint continuity?

Consider a function $f(x,y,z)$ such that $f:\mathcal{X}\times \mathcal{Y}\times \mathcal{Z}\rightarrow \mathbb{R}$ with $\mathcal{X}\subseteq \mathbb{R}$, $\mathcal{Y}\subseteq \mathbb{R}$ and ...
1
vote
0answers
40 views

Let $c : P(N) \rightarrow Z$ be the function defined by $c(X)= -1$ if $X$ is empty and $c(X)$ equals the least element in $X$ if $X$ is not empty.

Let $c : P(N) \to Z$ be the function defined by $c(X) = -1$ if $X$ is empty and $c(X)= \min(X)$ if $X$ is not empty. (a) What is $c(\{5,15,22\})$? (b) What is $c(\{x : x \in N$ and $x$ is an odd ...
0
votes
1answer
16 views

What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
2
votes
5answers
64 views

Pade approximant for the function $\sqrt{1+x}$

I'm doing the followiwng exercise: The objective is to obtain an approximation for the square root of any given number using the expression $$\sqrt{1+x}=f(x)\cdot\sqrt{1+g(x)}$$ where ...
0
votes
0answers
35 views

How to find the ordered pairs and range from the function $a: \{12,13,14,15,16,17,18,19,20\} \to\mathbb{Z}$

Let $a: \{12,13,14,15,16,17,18,19,20\} \to \mathbb{Z}$ be the function defined by $a(x)$ equals the largest prime number that divides $x$. a) Write down the set of ordered pairs which ...
3
votes
1answer
91 views

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation..

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation $$-1\leq f(x)\leq 1 $$ for all $-1\leq x\leq 1$, then the maximum value of $f'(x)$ is I think ...
7
votes
3answers
268 views

Function to describe teardrop shape

If I fill a plastic ziploc-shaped bag with water, the cross section profile should be sort of teardrop shaped (assuming we ignore the edge effects of the bag being sealed on the sides as well as the ...
6
votes
3answers
518 views

Different arrows in set theory: $\rightarrow$ and $\mapsto$ [duplicate]

Can someone explain the difference between symbols: $\rightarrow$ and $\mapsto$ Thanks.
0
votes
2answers
58 views

What does the function f: x ↦ y mean?

I am doing IGCSE Maths, and am having a few problems with function notation. I understand the form f(x). What does the form f: x ↦ y mean? Could you also give one or two examples? And, if possible, ...
3
votes
1answer
32 views

For the function $f(X) = x+ \frac 1x$, if maxima is found using second derivative test we get $x=1$ as the answer. But isn't $x = -1 $ the answer?

$$f(x) = x +\frac 1x$$ $$f'(x) = 1- x^{-2}=0 $$ $$\implies x= \pm 1$$ $$f''(x)= 2x^{-3}$$ $$f''(-1)<0$$ $$f''(1)>0$$ Therefore $x = 1 $ is the minimum. but $f(1)= 2 > f(-1) =-2.$ which ...
0
votes
0answers
12 views

Upper bound on the remainder of a polynomial (not taylor)

There are many ways of approximating a function with a polynomial, $\widehat{f}(x)\approx f(x)$. One way is the taylor polynomial. A nice property that goes along with the taylor polynomial is an ...