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

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4
votes
1answer
101 views

Iterated function?

$$f(n) = \frac{n}{\lg n}$$ $$g(n) = \min (i \ge 0: f^i(n)\le 2)$$ In other words, $g(n)$ is the number of times $f(n)$ needs to be iterated to reduce $n$ to 2 or less. What's a tight bound on ...
2
votes
4answers
131 views

How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?

Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional. I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
0
votes
0answers
184 views

Multiplication and Division of functions

Suppose that you have two continuous functions, $f(x)$ and $g(x)$. Suppose that you have numerical approximations for these functions, stored a vectors, $f^*$ and $g^*$. If I want to approximate ...
3
votes
1answer
45 views

Preimages of a function: Is the following proposition true or false?

Let $g: ℤ \times ℤ → ℤ \times ℤ$ be defined by $g(m,n) = (2m, m – n)$. Is the following proposition true or false? Justify your conclusion. For each $(s, t) ∈ ℤ \times ℤ$, there exists an $(m, n) ∈ ℤ ...
3
votes
2answers
42 views

$f(g(x))=x$ implies $f(x)=g^{-1}(x)$

Is it possible to find a necessary and sufficient condition to conclude when $$f(g(x))=x \implies f(x)=g^{-1}(x) \wedge f^{-1}(x)=g(x),$$ if both functions are well defined?
1
vote
2answers
91 views

Conditions for f(x,y) to coincide with some g(x) + h(y)

Given a function of two variables, say f(x,y), how to know/check whether it can be simplified into some g(x) + h(y)? Some property of some condition that condition that f(x,y) satisfies to know ...
5
votes
2answers
117 views

Inverse and derivative of a function [duplicate]

Find an example of an inverse function f(x) such that its derivative is the same as its inverse. I tried many different functions but non of them worked.
4
votes
2answers
197 views

Is $y=|x^3|$ a smooth function?

Is this a smooth function? $y=|x^3|$ The graph of this function has no sharp cuts or corners, so I think it is a smooth function but someone told me that it's not.
11
votes
2answers
711 views

$\cos(x)+\cos(x\sqrt{2})$ is not periodic

Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
0
votes
0answers
51 views

Help me prove the supremum property.

Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that $$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
1answer
56 views

is this function injective?

Is this function injective? $f:\mathbb{N}\to \mathbb{N\times N}$ defined as $f:n\to (n, n+1)$ $f(n_{1})=f(n_{2})\Rightarrow (n_{1},n_{1}+1)=(n_{2},n_{2}+1)\Rightarrow$ $n_{1}=n_{2} \wedge ...
30
votes
4answers
1k views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should ...
0
votes
1answer
94 views

Vector valued functions [closed]

What is a vector valued function ? Vectors are just $n-tuples$ , and then how are we able to describe geometrical shapes using them ? Is a vector function such that it outputs $n-tuples$ using other ...
0
votes
2answers
62 views

prove $f^{-1}(B)=A$

I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$ I want to prove that $f^{-1}(B)=A$. I just assume that here one is talking about ...
2
votes
2answers
54 views

Proving the coefficient of Power series is “0” always on given condition.

Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
0
votes
1answer
104 views

Proof: $f: A \to B$, if $f$ is bijection then $\forall a\in B \exists ! b\in A (f(b) = a) $

I musti proof the following: " let $f: A \to B$, if $f$ is injection then $\forall a\in B \exists ! b\in A (f(b) = a)$" proof: by contraddiction, therefore the negation of $\forall a\in B \exists ! ...
1
vote
2answers
36 views

Proving a condition on a cont-differentiable function on positive real numbers.

Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$. I am not getting how we will ...
2
votes
0answers
39 views

Inferring simplest method to convert bit array 1 to bit array 2.

Consider the set of all bit arrays of length $n$. Now consider the set of all 1-to-1 functions that map from this set to this set. Now select a single function out of the latter set. Is there any ...
19
votes
12answers
3k 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
84 views

Uniform probability density function

If I have uniform probability density function: $\pi(r)=1 \space \forall r\in [0,1] \space and \space 0 \space \forall r \notin [0,1]$ I just don't get it. If I integrate function f(x) from 0 to 0,5 ...
3
votes
0answers
56 views

What's the mathematical field called where functions create and delete functions?

Motivation In the field of modular, reconfigurable robotics there are some groups which use term rewriting, or specifically graph rewriting to describe the reconfiguration process of the modular ...
4
votes
5answers
684 views

Checking whether a polynomial of high degree is bijective or not.

Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false? I think is when we take ...
1
vote
0answers
24 views

Distinction between function types

Hi I'm am currently working on a question on Big O notation and to work out $O(n^k)$ of $f(n)$ you first need to know what type of function $f(n)$ is, polynomial, exponential, logarithmic. My question ...
1
vote
5answers
793 views

How do I find the image of the functions $y=2$ and $y = 2x - 6$?

The function is $y=2$, the domain is just 2? And the image of it? I don't think I quiet understand what the image of a function means, the domain is all values that it can assume, correct? Could you ...
0
votes
3answers
70 views

Simple onto functions

Find continuous and onto real functions between the following(if possible) $(0,1)\rightarrow(0,1]$ $(0,1]\rightarrow(0,1)$ $(0,1) \rightarrow(0,1)\cup(2,3)$ In 1 we can define a function like ...
3
votes
1answer
81 views

Formula for ' constant-power' across 3 sound sources (3-way DJ Crossfader)

First time poster here - thank you profusely in advance for any help you can provide! Intro/Context I need help to expand upon an existing mathematical approach to providing 'constant power' in ...
2
votes
1answer
74 views

Show that this type of function is surjective iff it's injective.

Here's a theorem that I think intuitively makes sense, but I was hoping to prove more rigorously: Theorem: Suppose $|A|=|B|=n$, where $n\in\mathbb{N}$. Consider the function $f:A \to B$. Then $f$ ...
1
vote
1answer
66 views

Triple Integral over a disk

How do I integrate $$z = \frac{1}{x^2+y^2+1}$$ over the region above the disk $x^2+y^2 \leq R^2$?
4
votes
1answer
137 views

How to express $\cos(\frac{x}{k})$ and $\sin(\frac{x}{k})$ in terms of $\cos(x)$ and $\sin(x)$, respectively?

How can we express $\cos(\frac{x}{k})$ ($k \in \mathbb{N}$) in terms of $\cos(x)$? And $\sin(\frac{x}{k})$ in terms of $\sin(x)$? Edit Maybe this another question helps. Is there a $T_n(x)$ ...
1
vote
2answers
106 views

Partial derivatives.

Suppose $$f(x+y, x^2 +xy + z^2) = 0.$$ Show that $$x + y = 2z\left(\frac{\partial z}{\partial y}-\frac{\partial z}{\partial x}\right).$$ Please help I don't know where to start!
0
votes
1answer
39 views

Prove that $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}$ is Lipschitz

I want to prove that a function $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a Lipschitz function. I proved that $|\phi(x_1,C_1) - \phi(x_2,C_1)| \leq A |x_1 - x_2|$ and $|\phi(C_2,y_1) - ...
1
vote
1answer
62 views

What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?

Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$. a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $ b) $D = \{y \in Y : ...
2
votes
2answers
289 views

Function that is Riemann-Stieltjes integrable but not Riemann integrable?

This is my first question, so please go easy on me :3 - I've searched, and I haven't found any questions that are particularly similar to this one. I'm reading Rudin's Principles of Mathematical ...
2
votes
2answers
137 views

Sketching a graph under certain condtions

I got a question like this, sketch the graph of a function that satisfies the following conditions, the domain is [0,oo]; the range is [4,oo]; the curve passes through [0,5]; while I was ...
1
vote
2answers
68 views

show that a function is bijective

If $A\approx B$ then $A^{C}\approx B^{C}$ where $B^{C}:=\{f|f:C\to B\}$ by ''$\approx$'' I mean equinumerous Proof: By hypothesis $A\approx B\Rightarrow \exists T:A\to B$ which is bijective and ...
1
vote
0answers
45 views

This is an easy question.Is this picture right?

In my text book about Fourier Analysis, $x$ ranges from $-\pi$ to $\pi$.I think when $x$ is from $-\pi$ to $0$, $\displaystyle \ln(2\sin\frac{x}{2})$ is meaningless. Why does the left part of the ...
4
votes
3answers
150 views

Looking for a differentiable function which behaves somewhat like $\min(x,1)$

Is there a differentiable function $f : [0,2] \rightarrow [0,1]$ such that $f(x) = 0$ iff $x=0$ and $f(x) = 1$ iff $x \in [1,2]$? What about $n$ times differentiable for any $n$, or infinitely ...
5
votes
3answers
422 views

Square integrable function that doesn't go to zero?

I'm reading through some elementary quantum mechanics textbooks and a few authors mention that there are functions that are "there exist pathological functions that are square-integrable but do not go ...
4
votes
3answers
86 views

$f(x)=\tanh(1+\tanh^{-1}(x))$ or $f:\tanh(x) \to \tanh(x+1)$ is a rational function?

This is (again) more a recreational/incidental question. Playing with iteration of functions I considered the function $$ f(x) = \tanh(1+\tanh^{-1}(x)) \tag1$$ such that $$ f : \tanh(x) \to ...
1
vote
2answers
31 views

Investigating functions - Lagranges mean value theorem

With the aid of Lagrange's formula prove the inequality : $ \frac{a-b}{a} \leq ln \frac{a}{b} \leq \frac{a-b}{b}$ for the condition $ 0 < b \leq a$ Please guide how to proceed for this.. ...
2
votes
6answers
233 views

Function Notation

due to our national cirriculum (the way in which it was taught in high school). We just said that f(x) means a function. Though I understand this isn't necessarily correct? In high school we used ...
2
votes
1answer
62 views

When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?

I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
3
votes
3answers
64 views

Find no. of points where $f$ and $g$ meet.

If $f(x)=x^2$ and $g(x)=x \sin x+ \cos x$ then (A) $f$ and $g$ agree at no points (B) $f$ and $g$ agree at exactly one point (C) $f$ and $g$ agree at exactly two points (A) $f$ ...
3
votes
1answer
723 views

Intermediate Value Property and Discontinuous Functions

This is a general question to which I need help finding a concrete example so that I may understand the concept/strategy better, and any help will be greatly appreciated. If given a function $F$ that ...
0
votes
1answer
60 views

under what conditions is f(A ∪B)=f(A) ∪f(B) and f(A∩B)=f(A)∩f(B)? [duplicate]

Does the function need to be bijective? I know for f(A∩B)=f(A)∩f(B) the function has to be injective, but what about the first equation?
-1
votes
1answer
62 views

What is the property of a function called that the value is the same given an input? [closed]

I was wondering given a function like $f(x) = x +5$ that for a given $x$, the value will be the same.
1
vote
1answer
51 views

Approximating Lipschitz funtion by $C^1$ function.

Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
1
vote
2answers
38 views

Homeomorphism $id_M:(M,\tau_d)\rightarrow(M,\tau_h)$

I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where $(M,d)$ is a metric space, ...
0
votes
1answer
42 views

percentages…

I have a sheet of plywood, say 10 sq. ft. I sell two pieces. Then, Jim bought a 5 sq ft piece while Joe bought a 2 sq. ft piece. The rest of the sheet of plywood is no good to me, so I want to ...
0
votes
2answers
41 views

Finding the derivative with functions inside, such as $g(x) = \dfrac{3x-1}{f(x)}$

With a question such as: $$g(x) = \dfrac{3x-1}{f(x)}$$ How does one approach finding the derivative, could the Chain Rule be used? The book, gives the answer as: $g'(x) =\dfrac {3f(x)-(3x+1)f'(x) ...