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

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0
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2answers
32 views

Name or closed form for given description of a function

I am looking for the name or nice explicit formula for the following function: I give you a positive integer N, and the function I want from you, say, f(X), subtracts X from N until the result is ...
-1
votes
4answers
70 views

Disprove the statement: If $g\circ f=I_X$then $f\circ g=I_Y$. [closed]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: Let $f : X \rightarrow Y$ and $g : Y \rightarrow X$ be functions. If $g\circ f=I_X$...
0
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1answer
30 views

Prove for some $n$, $f^{n+1}=f^n$ and that Y is bijective.

Are these sufficient to show what is being asked? If you could confirm or provide a more efficient way to do so I would greatly appreciate it. Let $X$ be a finite set and $f:X\rightarrow X$ be a ...
0
votes
1answer
23 views

How to prove that a function only defined for integers (or primes, or multiples of 3, etc.) have a certain derivative?

I would like to know how a function not continuously defined (for a lack of better words) could be proven to have a certain derivative (or, if the word "derivative" isn't appropriate, rate of growth). ...
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votes
0answers
14 views

How to substitute demand parameters from one function into another

Here are two inverse demand functions: $$ p_1= α_1-β_1q_1= 8−2q_1 $$ and $$p_2 = α_2-β_2q_2 = 4−\frac{1}{2q_2}$$ How can you get this:$$ gcs_1(q) = q(8−q)$$ and $$gcs_2(q) = \frac{q(16−q)}{4}$$ from ...
0
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0answers
77 views

Normal family theorem regarding meromorphic functions (Schiff, Joel)

I have a question regarding Theorem 3.3.1 from pages 76-77 in Joel Shiff's book Normal Families. The theorem is stated as such: $\textit{Let} \, \, \mathcal{F}$ $\textit{be a family of meromorphic ...
1
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1answer
23 views

Production functions total cost

Production function is: $f(L,M)=L^{1/2}M^{1/2}$. L is the number of units of labour, M of machines used. Cost of labour is 9 per unit, whereas the cost of machine is 81 per unit. Total cost of ...
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1answer
36 views

How to prove the following function is convex?

I was working on a problem and it reduced to show that $$f(a)=log\Big(\sum_{i=1}^{r}a^ix_i\Big)~~a>1, x_i>0$$ is convex. I have $$f^{\prime \prime}(a)=\frac{\partial^2f(a)}{\partial a^2}=\frac{[\...
2
votes
3answers
51 views

Would this proof be considered true.Proving a property of a operation

Ok so the operation [x] is defined to be equal to the integer such that it is $\leq x$ From this definition it holds that : $$ [x] \leq x $$ I need to prove that $$ [x+n] = [x] + n $$ My proof ...
3
votes
1answer
78 views

Let ${f_n}$ be a sequence of continuous real valued functions on $[0, \infty)$, then which of the following is/are true?

Let $\{f_n\}$ be a sequence of continuous real valued functions on $[0, \infty)$. Suppose $f_n(x)\to f(x) ~~~\forall x\in [0,\infty)$ and $f$ is integrable. Then $\int_{0}^{\infty} f_n(x)dx \to \...
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votes
1answer
14 views

How to determine if a function is quasiconcave or quasiconvex using calculus

I would like to know if there is a theorem which links the quasi concavity of a function to the sign of its second order derivative. For eg. we know a function is Concave in a given interval if it's ...
0
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1answer
22 views

Problem in equality of two functions.

First of all, my book states that two functions $f$ and $g$ are equal iff.: $Dom(f) = Dom(g)$ $Codom(f) = Codom(g)$ For each $x\in Dom(f)$ , $f(x) = g(x)$ and vice-versa. which is fairly ...
2
votes
2answers
38 views

Finding the function of a sine graph that has both translation and transformation

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ...
-1
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0answers
28 views

Kuratowski's representation for an ordered pair [closed]

Could anyone explain Kuratowski's representation for an ordered pair for me? I wish to know what 'adjoined with' means and how nested sets are made.
2
votes
1answer
38 views

Adding functions when one has undefined point [closed]

If two functions are defined as set of points and there's a point that is defined only in one function but not in the other, e.g. $A = \{(0,1)\}, B = \{(1,2)\}$. In function arithmetic, what would be $...
-5
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0answers
27 views

Complete the square [closed]

Research results for a clothing retailer show that 600 people will buy a sweater for 130\$. For every 1\$ increase in price, 25 fewer people will buy the sweater. The function that models the revenue ...
1
vote
0answers
29 views

Motivation behind substitutions in an integral 1

I was reading a textbook on Integration where I came across suggested substitutions for certain types of Integrals. These were as follows: $$$$ Integrals of the form $$\int\dfrac{dx}{(ax+b)\sqrt{...
0
votes
0answers
33 views

How is the Möbius function in boolean sets?

Although the text is a little long, the text is very simple so that're familiar with the matter. I did not understand two passages in the text. Could you help me show that: $I'= I \cup M$ and only ...
1
vote
2answers
41 views

Composition of a piecewise function and another function

I have this two functions. $f(x)=\arcsin \left(\dfrac{3-x}{3x-1} \right)$ and $g(x)=\begin{cases} 0 ;& |x| <\pi \\ \sin(2x);& |x| \ge \pi \end{cases}.$ I have to find $f \circ g$. I ...
0
votes
1answer
19 views

Exception to definition of a function.

My book gives me this definition of a function: A function $f$ is a special kind of relation,i.e $f\subset A\times B$,such that the following hold: for each $a\in A$ there exist $b \in B$ ...
1
vote
1answer
53 views

Functional inequality $f(x_1+x_2)\ge f(x_1)+f(x_2)$

Given a function $f$ on the interval $0\le x \le 1$. We know that this function is non-negative and $f(1)=1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1\ge 0, x_2 \ge 0$ and $x_1+x_2\...
0
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1answer
19 views

resolvable function

I need help. Show that the system of equations: $$y_1 + \cos(y_1y_2) = y_2*x_1 + 1$$ $$\sin(y_1) = x_2 + y_2$$ in an environment of $(x_1, x_2, y_1, y_2)=(0, −1, 0, 1)$ , $$\begin{pmatrix} y_1\\y_2 \...
5
votes
2answers
172 views

Integrate a periodic absolute value function

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
24
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4answers
8k 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?
3
votes
4answers
111 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
0
votes
0answers
21 views

Iterating a relation to find a function

I was playing around with a graphing calculator, trying to find approximations for inverses of $f(x)=x^5+x+1$. This cannot be expressed with radicals or the like, but I wanted to see how close I could ...
1
vote
2answers
49 views

Recursively counting divisors of a number

I want to make a recursive function f that counts all (not only prime) different divisors of a given natural number: $f(n): = |{a ∈ ℕ | ∃ b ∈ ℕ : a . b = n }| $ ; with $ f(0)=0 $ for example $ f(3) ...
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0answers
24 views

Prove a primitive recursive function

I'm supposed to prove that the division of integers (whole numbers) is primitive recursive. I know that the add, subs, mult are primitive recursive but I don't know if that helps, I'm not asking for ...
0
votes
0answers
20 views

Poll Ranking Formula

I am having a hard time making a ranking formula for a project I am working on. Here's a practical approach and I would really appreciate any help. Lets say we have $1000 to share among 5 people,...
0
votes
1answer
35 views

How can I find the largest possible subset A of $\mathbb{R}$?

So I have this equation $$f(x)= \frac{x^2}{(x-2)(x+3)}$$ and I need to find the largest possible subset $A$ of $\mathbb{R}$ that could form the domain of a function. Can anybody help me? I really don'...
1
vote
1answer
25 views

The function $f:X\rightarrow X$ is defined in such a way: $f(x)=x-\frac1x, f^{(1)}(x)=f(x), f^{(n)}(x)=f\left(f^{(n-1)}(x)\right)$

Let the set $X=\mathbb Q / \{-1;0;1\}$. The function $f:X\rightarrow X$ is defined in such a way: $$f(x)=x-\frac1x, f^{(1)}(x)=f(x), f^{(n)}(x)=f\left(f^{(n-1)}(x)\right), n\in \mathbb N.$$ Is ...
12
votes
2answers
362 views

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor 2^...
2
votes
2answers
27 views

Complex function $f$ is either constant or unbounded, but maximum value still does exist even if $f$ is not constant?

In Complex Variables and Applications, Brown & Churchill (9th edition), I stumbled upon a chapter which got me somewhat confused. On page 175 of the book, there is the theorem, which states the ...
1
vote
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 ...
0
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1answer
46 views

Simplification of this function

$f(x)= ((1- 4x^2)^{1/2} - 2(3)^{1/2}x)/((3-12x^2)^{1/2} + 2x)$ Find range when $x$ belongs to $(-(3)^1/2 /4 ,1/2)$ I have to find the range of this function , I have simplified this expression , I ...
2
votes
2answers
40 views

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \...
2
votes
2answers
38 views

How do I find the domain/range of functions algebraically?

I've been having trouble when trying to find the domain/range of functions algebraically. Here is an example: $P(x)=\frac{1}{3+\sqrt{x+1}}$ Finding the domain: $x+1\ge0$ $x\ge-1$ Therefore, $x \...
0
votes
1answer
15 views

Finding Domain of a Function with a natural logarithm at the denominator of the fraction

I have the function: $y = f(x) = \frac{x}{\ln x}$ The function is undefined for the conditions: a denominator of a fraction being zero. a logarithm being negative or equal to zero. Hence, is ...
0
votes
3answers
87 views

Find the minimum of the value $n$ such that $(1-0.03)^n<0.03$

How can I find the smallest positive integer $n$ such that $$(1-0.03)^n<0.03$$ without the help of a computer?
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1answer
28 views

Show that every local homeomorphism is continuous and open therefore bijective local homeomorphism is a homeomorphism

Follow up on another question I asked recently: Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism Definition: Let $(X, \mathcal{...
0
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2answers
46 views

Show that cuts are preserved under homeomorphism

Let $(X, \mathcal{T})$ be a topological space, assume that $X$ has no proper (not $X$ or $\varnothing$) clopen subset. Definition: A point $p \in X$ is a cut if $X \setminus\{p\}$ has a proper ...
2
votes
1answer
75 views

If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such that $f(x)=c$ for all values of $x$ in the domain of $f$?

Here is a problem from Rusczyk-Crawford's Art of Problems Solving: Intermediate Algebra textbook (Chapter 2 Review, problem 2.30). If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such ...
5
votes
2answers
56 views

Why isn't f(x^2) a horizontal stretch of f(x) by a factor of “1/x”?

I know this question seems silly, but it came to mind while reading about transforming functions. Is the statement "y=f(kx) results from scaling the graph of y=f(x) horizontally by a factor of 1/k" ...
1
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1answer
31 views

Repeating/“Periodic” Derivatives? [duplicate]

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to ...
0
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2answers
37 views

Cardinality of the Domain vs Codomain in Surjective (non-injective) & Injective (non-surjective) functions

I'm a student in college just beginning to study the basics of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
1
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1answer
30 views

is $G(x) = |x-2|$ one to one OR onto? [closed]

Please help me determine if $G(x) = |x-2|$ is one to one or onto.
0
votes
1answer
37 views

Proof of a Surjective Function

I've run into a question in my textbook and I'm not sure if I understand fully the answer from the solution manual. Here is the question: Problem: Suppose that $f: A \rightarrow B$ is any function. ...
-1
votes
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. ...
2
votes
2answers
55 views

How to show that the following function is bijective?

If we have the function $c : \mathbb{N}^2 \rightarrow \mathbb{N} : (x,y) \rightarrow 2^x \cdot (2y+1) -1 $ how to show that this function is bijective? So I thought the easiest way is to show that is ...
0
votes
0answers
20 views

Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...