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

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3
votes
1answer
387 views

Creating a bijection from $(a,b)$ to $\mathbb R$ that is visually compelling

Given this "proof without words" from MO I am trying to: a) find a function that behaves like the function shown, on an open interval - say $(-1/2,1/2)$ b) find some intuition for why/how this ...
0
votes
2answers
46 views

Function of series

If a function $F(x) = \sum_i a_i/(x - y_i)$. Is it possible to simplify F(x) so that the repeated sum for each value of x can be avoided ?
0
votes
1answer
75 views

How can I minimize the following function

I have a simple function and want to minimize it using any method with precision of about 0.01. The domain of x is [-2,2]. The function is a simple $f(x) = x^2$ plus a triangle in the form of ...
2
votes
1answer
607 views

One-to-one correspondence between infinite sets

What does it mean to have one-to-one correspondence between infinite sets. How would you solve them?
-1
votes
2answers
174 views

Pigeonhole Principle

Explain the following using Pigeonhole Principle is it is true: 1) If we choose 10 points in a $3 x 3$ inch square, there must be two points of the 10 which are at distance less than or equal to ...
3
votes
3answers
2k views

Counting 1:1 and onto functions

I'm faced with the following questions: 1) How many functions are there from a set of size 3 to a set of size 5? How many of them are 1-to-1? 2) How many functions are there from a set of ...
0
votes
3answers
71 views

If $f(x)$ is $1$–$1$, must $g(f(x))$ be $1$–$1$?

Say $f$ is $1$–$1$, and both $f$ and $g$ are from $\mathbb N \to \mathbb N$. Does $g$ have to be $1$–$1$? What about $f \circ g$ ?
8
votes
2answers
347 views

Prove the following property of $f(x)$?

Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$ Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
0
votes
2answers
62 views

Determination of Functions, 1:1, and Inverse

For the following relations, I need to answer: 1) Is it a function? If not, explain why and stop. Otherwise, 2) What are its domain and image, 3) Is the function 1:1. If not, explain why and stop. ...
1
vote
1answer
115 views

Composition of step function with linear one.

Suppose that $s$ is a step function, and $f(x)=mx+k$. How to prove that $s(f(x))$ is step function also? Edit: Definition of step function: Function $s$ defined on $[a,b]$ will called a step ...
4
votes
7answers
146 views

How should I understand $f^{-1}(E):=\{x\in A:f(x)\in E\}$?

I understand the concept, but I still can't figure out how to read the notation: $$f^{-1}(E):=\{x\in A:f(x)\in E\}$$ I understood the concept due to the examples, not with the notation. Can someone ...
2
votes
2answers
89 views

When is $f=f^{-1}$ for $f(x)=x^3+14x-14$?

The function $f(x)=x^3+14x−14$ is a monotonically increasing function, hence it is injective (one-to-one), so its inverse function exists and is well defined. How many points of intersection are there ...
2
votes
2answers
1k views

What is the purpose of the inverse image?

I'm reading Bartle and Sherbert: Introduction to Real Analysis. The author introduces the definitions of direct and inverse images: Let $f:A\rightarrow B$ be a function with domain $D(f)=A$ and ...
7
votes
3answers
347 views

Finite at every point but unbounded on every interval

Is is possible that a function $f$ is finite at every point but unbounded on every interval? What if f is measurable?
3
votes
1answer
77 views

Does $f(x)$ exist such that $f(x)$ can't be integrated but $f'(x)$ can?

I'm looking for an example of a function (if such a function exists) that cannot be integrated, but its derivative can. Also, Does such a positive function exist, such that its co-domain is always ...
2
votes
2answers
198 views

Prove that f(X) is constant.

Now I have seen a lot of answers around here which seem to be good enough. Problem is, our teacher asked us to prove it his way. Suppose we know that $$|u(x)−u(y)|≤(x−y)^2$$ Prove, by adding and ...
0
votes
2answers
33 views

Curves of functions

Please, how can we draw the curve of $ f(x) $ if we have the curve of $ g(x) $ where : $ f(x) = x-\sqrt{|x^2-4|} $ and $ g(x) = x+\sqrt{|x^2-4|} $ Thank you
0
votes
3answers
2k views

If $f$ and $g$ are onto, then show that $g(f(x))$ is also onto [duplicate]

$f$ is a function from $A$ to $B$ and $g$ is a function from $B$ to $C$, show that if $f$ and $g$ are both onto, then $g\circ f$ is also onto.
12
votes
4answers
2k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
0
votes
3answers
850 views

How to determine the difference Onto vs One-to-one?

I have no clue how to solve for this. How can I prove if a function is either onto or one-to-one in the following example? Let $S$ be the set of all strings of 0's and 1's, and define $D:S \to ...
2
votes
3answers
60 views

Determining whether or not $f$ is one to one

I am not sure where to start or how to find a solution. How can I determine this is a one to one function: $f(x) = x + \frac1{x - 1}$, for all real numbers $x \ne 1$.
2
votes
1answer
52 views

Differentiability of first derivative of a function

If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples? From Darboux ...
4
votes
2answers
207 views

Counting the number of functions

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
1
vote
1answer
155 views

Injectivity of a Function [closed]

Sorry for confusion. I am in the process of solving a functional equation, I need to show injectivity. (By the way i know that it is injective, I'm trying to prove it to myself). Putting $f(x)=f(y)$ ...
5
votes
2answers
130 views

Showing $\{x\} + \{\frac{1}{x}\} \lt 1.5$ and other problems.

For any real number $x$, let $[x]$ be the greatest integer not exceeding $x$. We also define $\{x\}=x-[x]$. We now define the function: $f(x)=\{x\}+\{\frac{1}{x}\}$. (a) Prove that $f(x)<1.5$ for ...
1
vote
1answer
145 views

Does Euler's homogenous function theorem hold for functions homogenous in some of its independent variables?

All proofs of Euler's homogenous functions theorem I've come across seem to assume the function is homogenous in all its independent variables. But does the theorem also hold in the case where the ...
1
vote
1answer
27 views

Functional form

Is there an analytic form of a function that satisfies the following: $$\begin{align*} f:\mathbb N &\to \mathbb N\\ 1&\mapsto 1\\ 2&\mapsto 2\\ 3&\mapsto 2\\ 4&\mapsto 3\\ ...
6
votes
0answers
305 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
4
votes
4answers
2k views

Show that function is strictly monotone increasing

I want to show that $$ f(x)=\dfrac{x-\sin(x)}{1-\cos(x)} $$ is strictly increasing in $(0,2 \pi) $. Unforunately, this is not that easy for me , as the derivative is not very manageable and ...
1
vote
3answers
533 views

Determine if function is well defined

I am having difficulties determining if the following function is well defined: On certain computers the integer data type goes from $-2, 147, 483, 648$ through $2, 147, 483, 647$. Let S be the set ...
0
votes
0answers
15 views

Functions defined on General sets [duplicate]

I am learning how to determined whether a function is well defined. I am doing so by relying on two disticnt reasons that show a not well defined function: (1) There is no y that satisfies the given ...
3
votes
2answers
99 views

Qualifying exam question concerning roots of equation

Consider a polynomial $f_n(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ with integer coefficients. If $f_n(x_i)=19$ for $5$ distinct $x_i \in \mathbb{Z}$, find the number of distinct integer solutions for ...
1
vote
1answer
105 views

Quick equivalence class clarification question

A quick clarification question, what is an equivalence class of a function? For example if you have an identity function on all integers $I_{Z}$, what would $[I_{Z}]$ = ? I know that when you have a ...
5
votes
2answers
524 views

Sets, functions and relations problem

Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes: Question: Context: Let $A$ and $B$ be subsets ...
0
votes
1answer
340 views

Discrete Math functions and sets

Let $A$ and $B$ be subsets of $\Bbb{Z}$, and let $F = \{f : A\to B\}$. Define a relation $R$ on $F$ by: for any $f,g\in F$, $fRg$ if and only if $f - g$ is a constant function; that is, there is a ...
4
votes
1answer
322 views

Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$

Show that $f^{-1}(A\cup B) = f^{-1}(A)\cup f^{-1}(B)$ but not necessarily $f^{-1}(A\cap B)=f^{-1}(A)\cap f^{-1}(B)$. Let $S=A\cup B$ I know that $f^{-1}(S)=\{x:f(x)\in S\}$ assuming that that $f$ ...
0
votes
2answers
461 views

Product of Odd Functions

So I'm working on a mini-project for my intro proof writing course and we're given the following that I'm a little hung up on. Consider $V$ to be a known vector space and functions $f$ and $g$ such ...
0
votes
2answers
324 views

Function and equivalence relations question

Let A and B be subsets of the set Z of all integers, and let F denote the set of all functions f : A to B. Define a relation R on F by: for any f,g element of F, fRg if and only if f - g is a ...
4
votes
3answers
80 views

Finding a parameter of a function

I have a function: $f(x)=-\frac{4x^{3}+4x^{2}+ax-18}{2x+3}$ which has only one point of intersection with the $x$-axis. How can i find the value of $a$? I tried polynomial division and discriminant, ...
2
votes
2answers
915 views

Proving two functions are equal

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
1
vote
1answer
118 views

For which numbers $c$ is there a number $x$ such that $f(cx)=f(x)$?

This is one exercise in Spivak's book that is bugging me for a while, first I thought that $c=1$, but there's a hint: There are a lot more than you might think at first glance. And here I'm ...
1
vote
1answer
79 views

2 times differentiable functions with compact support

I know there are bump functions which are infinitely differentiable (smooth) and compactly supported. For example, this function $$ \phi(x) = \begin{cases} \exp(- \frac{1}{1 - x^2}), \quad |x| < ...
1
vote
1answer
70 views

Proving $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$.

Let $\mathscr F$ denote the set of all functions from $\{1,2,3\}$ to $\{1,2,3\}$. Prove or disprove. (a) $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$. (b) $\forall f\in ...
0
votes
3answers
122 views

Function mapping

If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$? It has been established that this is $m^n$. How many of these are one-to-one? I think this means that each ...
1
vote
2answers
487 views

Find the number of distinct equivalence classes $[f]$ of $R$.

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Define a relation $R$ on $\mathscr F$ by : for any $f,g ...
2
votes
1answer
80 views

Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
0
votes
2answers
49 views

Proof of functional equation

I have a function $$ g_N(x) =\frac{1}{x}+ \sum_{n=1}^\infty \Big(\frac{1}{x+n}+\frac{1}{x-n}\Big) $$ How can I prove that $$g_N(\frac{x}{2})+g_N(\frac{x+1}{2}) = 2g_N(x)\;?$$
2
votes
4answers
62 views

Equality proof of function $g_N(x) = \sum_{n=-N}^N \frac{1}{x+n}$

I have a function $$ g_N(x) = \sum_{n=-N}^N \frac{1}{x+n} $$ How can I prove that this function is odd, thus $ g_N(-x) = -g_N(x)$ ?
0
votes
1answer
55 views

Central difference discrete

If they say that $f_i$ is the central difference discrete of $f(t,x)$ in the point $(t_n,x_i)$. What do they mean by this?
1
vote
2answers
221 views

Prove the following ceiling and floor identities?

Could someone help me prove these identities? I'm completely lost: $$\begin{align*} &(1)\quad \left\lceil \frac{\left\lceil \frac{x}{a} \right\rceil} {b}\right\rceil = \left\lceil ...