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

learn more… | top users | synonyms (1)

0
votes
2answers
40 views

Proving $f(n)=100n+5 \neq \Omega(n^2)$

I have to prove that: $$f(n)=100n+5 \neq \Omega(n^2)$$ What I tried: let's assume that $f(n)=100(n)+5= \Omega(n^2)$. Thus, there must exist some positive constant $c$ and $n_0$ such that, $$0 \leq ...
1
vote
2answers
56 views

When to rationalize to repair continuity, and why does it work?

I was working on a question out a GRE math prep book: "Find the inverse of $f(x) = \frac{x}{1-x^2}$ that works for all $x \in \mathbb{R}$ where $f$ is defined over $(-1,1)$" (works meaning is well ...
0
votes
1answer
36 views

Prove any function can be written as a composition between an injective and a surjective function.

Given an arbitrary function $f:A\rightarrow B$, write it as a composition between an injective and a surjective function, respectively.
0
votes
2answers
1k views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ^{x-1}$...
9
votes
2answers
193 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
2
votes
1answer
35 views

Finding the equation of a polynomial

A quadratic function with a minimum of 5 has zeros at -4 and 2, find the equation of this function. This is impossible, correct?
0
votes
1answer
28 views

All linear functions are homogeneous of degree one?

I was looking through the Wikipedia page of "Homogeneous functions" and it stated that any linear function that maps V onto W is homogeneous of degree one. However, when I try to apply the definition ...
2
votes
1answer
108 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
0
votes
3answers
46 views

How do I find Big O notation for this function?

How do I find Big O notation for this function? $$ n^4+100\cdot(n^2)+50 $$ In the book I am following, I got the following solution: $n^4+100(n^2)+50 \leq 2(n^4) \ \forall \ n \geq 11$ $n^4+100(n^2)+...
0
votes
1answer
25 views

Question about continuous onto maps of homeomorphic spaces.

If $f:(A,T) \rightarrow (B,T_1)$ is continuous and onto, and $$(A,T) \cong (C,T_2) \land (B,T_1) \cong (D, T_3)$$ $$\Rightarrow \exists g: (C,T_2) \rightarrow (D,T_3)$$ that is continuous and onto.
9
votes
4answers
161 views

Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$

Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions? $$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
7
votes
3answers
2k views

Commutativity of iterated limits

The following is a weird result I've obtained with iterated limits. There must be a flaw somewhere in someone's reasoning but I can't discover what it is. The problem is that, in general, iterated ...
0
votes
1answer
72 views

What's the order of operations when dealing with function composition?

Given $f:[0,1]\rightarrow \mathbb{R}$ and $g:[0,1]\rightarrow [0,1]$, $g(x)=x^2$. Which of the two equalities is true? 1)$f^2(x^2)=f^2(g(x))=(f^2\circ g)(x)$; 2)$f^2(x^2)=f(x^2)\cdot f(x^2)=f(g(x))...
1
vote
0answers
45 views

Sigmoid function with fixed bounds and variable steepness [partially solved]

(see edits below with attempts made in the meanwhile after posting the question) Problem I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a ...
0
votes
1answer
429 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the $y$...
0
votes
0answers
40 views

Prove that $\{f^{-1}(B_i)|i\in I\}$ is a partition of X.

Could someone confirm what I have shown is sufficient in proving what was asked? I have no other way of checking my proofs and any help would be appreciated. Thank you for your time. Let $f:X\...
0
votes
1answer
22 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
1
vote
1answer
17 views

Function exercise check-up

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
-1
votes
2answers
29 views

Marginal revenue of a monopolist [closed]

A monopolist faces a demand function $Q=4000(p+7)^{-2}$. If she charges a price of p, her marginal revenue will be: (a) $p/2+ 7$ (b) $2p+ 3.50$ (c) $p/2-7/2$ (d) $-2(p+7)^{-3}$ Correct answer is ...
3
votes
2answers
158 views

Is it ok to use Kronecker delta function to find if one of its variables belongs to a half open interval?

Kronecker "delta" function is generally defined as $\delta(i,j)=1$ if $i$ is equal to $ j$, otherwise $0$. How about if $j$ is not an integer? I mean let $j$ is a half open interval defined as $j=(0,...
2
votes
3answers
72 views

How to find range of $\frac{\sqrt{1+2x^2}}{1+x^2}$?

How to find range of $$\frac{\sqrt{1+2x^2}}{1+x^2}$$ ? I tried put it equal to $y$ and squaring but I'm getting $4$th degree equation.
4
votes
4answers
212 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
1
vote
1answer
39 views

Find function for graph

I would like to find a function for the following graph: I have drawn the graph myself, so not all subtle bends are to be replicated. I have noted the important points the graph should have in the ...
0
votes
0answers
14 views

Notation for the index of minimum value of several variables

Assume we have several variables of the form $d_c$ which namely can be $d_1$, $d_2$, ..., $d_n$. I want to use mathematical notation to show for which index $c$ the value of $d_c$ is minimal for all ...
1
vote
4answers
61 views

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
-1
votes
0answers
26 views

square integrable function?

If I want to find out for which $\alpha$ the function $f:B_1(0)\to\mathbb{R}$, $f(x)=x|x|^\alpha$ is in $L^2(B_1(0))$, where $B_1(0)\subseteq \mathbb{R}^n$, can I do something like this: $$\int_{B_1(...
0
votes
2answers
38 views

How to antidifferentiate a function not applicable to basic antidifferentiation rules? [closed]

For example, how would one go about finding $\int (\pi(x)) dx $? Is there a certain technique or formula? That is, how does one antidifferentiate a function without using integration rules? How does ...
3
votes
2answers
63 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
3
votes
1answer
125 views

How many possible functions?

Take $f:\{1,2,3,4,5,6,7\}$ to $\{0,1,2,3,4\}$ How many such functions satisfy the cardinality of the pre-image of the set $\{3\}$ is equal to $3$. I thought it would be $35$, i.e :$7\choose{3}$ ...
0
votes
3answers
56 views

Inverse of $f(x) = 2x^2+8x+13?$

How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far: $y = 2x^2+8x+13$ $x = 2y^2+8y+13$ $x-13 = 2y^2+8y$ $x-13=y(y+8)$ This is where I got stuck. To be clear, I want ...
0
votes
1answer
33 views

Normal vector on a plot

Do a sketch of $f$ with the equation $f(x,y)=0$. Give in all non singular points of the curve a normal vector. $f(x,y)=x^{3}-x-y$ How can I do this thing with normal vector? I know that singular ...
1
vote
0answers
17 views

How do I solve this equation $f(x, y) = x - y^3 + y$ local for $h(x)=y$?

How do I solve this equation $f(x, y) = x - y^3 + y$ local for $h(x)=y$? $y^3+y=x$ What next?
1
vote
0answers
20 views

Calculate Density of Values in Cellular Automata

I am working with a special cellular automata that uses hexagonal cells rather than square cells, a hexagonal grid, rather than a square grid, and the set of complex numbers, rather than a finite set, ...
0
votes
2answers
31 views

Determine a and b so that function is continious

$$ g(t)= \begin{cases} 2t^2 ;& t<-1 \\ at ;&-1<t<1 \\ bt-\frac 12 ;&t>1 \end{cases} $$ How can I determine $a$ and $b$ so this function $g$ is continuous at whole $\mathbb R$. ...
10
votes
3answers
1k views

A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
-1
votes
3answers
74 views

Domain of the function $\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$

What is the domain of $$\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$$ the answer is $(-\infty,\infty)$. Now the polynomial has degree $12$. Also it's continuously increasing from $1$. So I thought there ...
0
votes
1answer
29 views

Determining if a function is onto

If our range such as in the question below is all the real numbers excluding $0$, to determine if a function is onto we must ask if all real numbers excluding $0$ can be mapped to at least one value ...
3
votes
1answer
253 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
-1
votes
0answers
14 views

Resolution function explicity [on hold]

Examine where the equation $f(x,y)=0$ locally by $y=h(x)$ can be resolved. Calculate in all these places $h'(x)$ by implicit differentiation. Enter the resolution function(s) $h(x)$ explicitly if this ...
0
votes
6answers
91 views

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$?

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$ ? All I could till now : LHS =$2\sin{x}+\cos{x}$ Since, $āˆ’\sqrt{5} \leq 2\sin{x}+\cos{x} \leq \sqrt{5}$ So a ...
-3
votes
1answer
47 views

Determine function is onto or not? [closed]

A function is defined as $$f(x) = \frac{e^{x^2}-e^{-x^2}}{e^{x^2}+e^{-x^2}}$$ f is from $\mathbb R\to \mathbb R$. check if function is surjective or not, injective nature of function can be proved ...
0
votes
1answer
32 views

Is the function $f: n \in N^{*} \longrightarrow |D(n)| \in N^{*}$ injective/surjective?

Is the following function injective/surjective? $$f: n \in N^{*} \longrightarrow |D(n)| \in N^*$$ (where $D(n)$ is the set of all the divisors of $n$). My attempt: It is NOT injective because $f(2) ...
4
votes
0answers
61 views

Prove that $f$ is invertible

Did I show enough to prove $f$ is invertible? Alternatively is there a more efficient way to do so? Thanks in advance for any help. Let $f : X \rightarrow Y $a nd $g : Y \rightarrow X$ be ...
5
votes
3answers
2k views

What does the “closed over”/“closed under” terminology mean exactly and where did it come from?

I've been trying to teach my partner some set theory, and I got thrown for a loop while trying to give her a precise definition of some basic terminology. So we've heard of a set being described as "...
2
votes
2answers
37 views

A question on mapping inside unit disc

For an analytic function $f:\bar Dā†’\bar D$ where $\bar D$ is the closed unit disc centered at origin.Suppose $\bar D=D\cup$$\delta D$, where $\delta D$ is the boundary of open disc $D$ and $f$ is onto,...
0
votes
2answers
51 views

Determining if a power set is one to one or onto.

Let $P$ be the power set of $\{a,b,c\}$. A function $f: P \to \mathbb{Z}$; the set of integers, follows: For $A$ in $P$, $f(A)=$the number of elements in $A$. Is $f$ one-to-one? Explain. Is $f$ onto?...
2
votes
1answer
58 views

What are all the different classes of functions upon real numbers and what do they mean, exactly? [closed]

I have been hearing terms like "piecewise C1", "continuous", "linear", "piecewise constant", "trigonometric", "logarithmic", "exponential", "elementary", etc. functions for many years. I know what ...
2
votes
0answers
26 views

2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
2
votes
2answers
56 views

If $C\cap D=\emptyset$ Prove that $f^c(C)\cap f^c(D)=\emptyset$

Is this the proper way to go about proving this? By showing $C\cap D$=$f^c(C)\cap f^c(D)=\emptyset$? Any feedback would be greatly appreciated. I don't have any other way of getting feedback for my ...
0
votes
2answers
44 views

How to find minimum and maximum of function?

Given: the function $$f\left(x,y\right)=4x^{2} +3y^{2} -5x$$ Find the $x$ values of the minimum and of the maximum on the set: $$ \left\{ \left(x,y\right)\in \mathbb R ^{2}: x^{2}+y^{2}=9 \right\}...