Elementary questions about functions, notation, properties, and operations such as function composition.
2
votes
0answers
27 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 ...
16
votes
9answers
1k 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
40 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
41 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 ...
0
votes
0answers
19 views
Function not returning a vector?
I've got a couple scripts I'm running but for some reason my y variable is a scalar and not a vector. Can anyone let me know why this is? Thanks.
For those interested, I am trying to determine a ...
4
votes
5answers
156 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
17 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
76 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
48 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 ...
2
votes
1answer
24 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 ...
0
votes
0answers
19 views
Formulating math problem with rounding / discrete step
I have this problem, that can easily be solved by simulation or numerical optimization, but I wonder how to write it as a mathematical problem? It's two pricing schemes, one cost is evaluated at ...
2
votes
1answer
26 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
39 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
62 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
77 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
33 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
50 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 : ...
1
vote
2answers
69 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
33 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 ...
0
votes
2answers
44 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
40 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 ...
0
votes
0answers
54 views
A question on a function
Let $f: X^2 \to I=[0,1]$ be a continuous function and $M$ be any
subset of $X$. Then is the folowing claim right?
The function $f_M$ from $X$ to $I^M$ with the
compact-open topology is ...
4
votes
3answers
88 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 ...
4
votes
1answer
66 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
48 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 $$ f : \tanh(x) \to ...
1
vote
2answers
16 views
Investigating functions - Lagranges mean value theorem
With the aid of Lagrange's formula prove the inequality :
$ \frac{a-b}{a} \leq ln \frac{a}{b} \leq \frac{a-b}{b}$
for the condition $ 0 < b \leq a$
Please guide how to proceed for this..
...
2
votes
6answers
95 views
Function Notation
due to our national cirriculum (the way in which it was taught in high school). We just said that f(x) means a function. Though I understand this isn't necessarily correct? In high school we used ...
2
votes
1answer
33 views
When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?
I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
3
votes
3answers
49 views
Find no. of points where $f$ and $g$ meet.
If $f(x)=x^2$ and $g(x)=x \sin x+ \cos x$ then
(A) $f$ and $g$ agree at no points
(B) $f$ and $g$ agree at exactly one point
(C) $f$ and $g$ agree at exactly two points
(A) $f$ ...
1
vote
1answer
52 views
Intermediate Value Property and Discontinuous Functions
This is a general question to which I need help finding a concrete example so that I may understand the concept/strategy better, and any help will be greatly appreciated.
If given a function $F$ that ...
0
votes
1answer
40 views
under what conditions is f(A ∪B)=f(A) ∪f(B) and f(A∩B)=f(A)∩f(B)? [duplicate]
Does the function need to be bijective? I know for f(A∩B)=f(A)∩f(B) the function has to be injective, but what about the first equation?
0
votes
1answer
49 views
What is the property of a function called that the value is the same given an input?
I was wondering given a function like $f(x) = x +5$ that for a given $x$, the value will be the same.
0
votes
0answers
15 views
Is this function a modular function?
Is the following function a modular function?
$$\xi(x) _q=\sum_{k=1}^q \frac{1}{2q}( exp(\frac{-i2\pi (k-1)x}{q})+exp(\frac{+i2\pi (k-1)x}{q}) )$$
with $x\in\Bbb{R}$ and $q\in\Bbb{N}$ and $q\neq0$.
...
1
vote
1answer
25 views
Approximating Lipschitz funtion by $C^1$ function.
Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
1
vote
2answers
28 views
Homeomorphism $id_M:(M,\tau_d)\rightarrow(M,\tau_h)$
I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where
$(M,d)$ is a metric space, ...
0
votes
1answer
25 views
percentages…
I have a sheet of plywood, say 10 sq. ft. I sell two pieces. Then, Jim bought a 5 sq ft piece while Joe bought a 2 sq. ft piece. The rest of the sheet of plywood is no good to me, so I want to ...
0
votes
2answers
34 views
Finding the derivative with functions inside, such as $g(x) = \dfrac{3x-1}{f(x)}$
With a question such as:
$$g(x) = \dfrac{3x-1}{f(x)}$$
How does one approach finding the derivative, could the Chain Rule be used?
The book, gives the answer as:
$g'(x) =\dfrac {3f(x)-(3x+1)f'(x) ...
1
vote
1answer
45 views
Proof involving functions.
Consider two functions $f\colon A \to B$ and $g\colon B \to C$. How can I prove the following?
If $f$ and $g$ are one-to-one, then the composition function $g \circ
f$ is one-to-one.
If $f$ and ...
4
votes
1answer
47 views
Suppose that the function f(x)
Suppose that a function $f(x)$ defined on $[0,1]$ satisfies
$f(1/n)\to 0$ as $n\to\infty$. Can we say that $f(x)\to 0$ as $x\to 0^+$ if $f$ is continuous on $[0,1]$ ? and again
is it true $f(x)\to ...
-1
votes
0answers
23 views
Formula for the sound pressure of a pure tone [closed]
What is the formula for the sound pressure of a pure tone of 500Hz, ex- pressed as a function of time?
0
votes
0answers
52 views
How to find the range or domain of a function?
This is a general question I'm asking, I really need it explained. Here's an example of what I mean:
The functions $f$ and $g$ are defined by
$f( x)= x^3 + 1$, $0 ≤ x ≤ 3$
$g(x)= x + 5$, ...
1
vote
6answers
79 views
A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$?
What is the range of the function
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$
0
votes
1answer
48 views
Additive maps modulo $1$ - what do they look like?
Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$).
A map $f:\ T \to T$ is ...
1
vote
2answers
75 views
Show that equation has no solution in $(0,2\pi)$
Hi I want to show that the equation $2=2 \cos(x)+x \sin(x) $ has no solution in $(0,2 \pi)$. Since it is algebraically impossible to solve this equation for $x$ I wanted to ask you whether one of you ...
0
votes
1answer
30 views
Special type of convexity
Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies
$$
f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
6
votes
2answers
116 views
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
0
votes
2answers
39 views
Show uniform convergence of indefinite function series
How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5]
I thought about using the Weierstrass ...
1
vote
2answers
39 views
Find a polar representation for a curve.
I have the following curve:
$(x^2 + y^2)^2 - 4x(x^2 + y^2) = 4y^2$ and I have to find its polar representation.
I don't know how. I'd like to get help .. thanks in advance.
1
vote
1answer
36 views
composite function with conditional IF
I've been wrapping my head around my Computer and Logic Essentials class, I can do most composite functions, however there is one question that I'm confused with.
It has an if statement inside it:
...
