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

learn more… | top users | synonyms (1)

1
vote
2answers
28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
0
votes
1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
1
vote
1answer
17 views

Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where ...
1
vote
1answer
39 views

Expanding function into Maclaurin series

How to expand that function below into Maclaurin series? Is it even possible? $$ f(x)=x^2+\ln\left(\frac{2x-3}{5-3x}\right) $$ I know that expanding into Maclaurin series requires function class ...
-2
votes
1answer
30 views

A Problem Distribution Function

If I have a probability density function like this $w(x) = 1 - |x| $if $|x| \leq 1$ or $ w(x)=0$ if $|x|\geq 1$, what's the value of the distribution function F(x)? I mean that I calculated ...
5
votes
0answers
40 views

Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
0
votes
1answer
24 views

Manually plotting some particular graphs

How to plot graphs like these manually: 1) $f(x)=\ln(1+x^2)$ 2) $f(x)=\frac8{2+x^2}$ 3) $f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}+\frac{\cos x}{\sqrt{1+\cot^{2}x}}$ I have no idea how to plot the ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
0
votes
1answer
15 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
-4
votes
2answers
40 views

No. of surjections [on hold]

Find the number of surjections from a $3$-element set to a $2$-element set. Find a formula for the number of surjections from $ℙ_{k+1}→ℙ_k.$ Find a formula for the number of surjections from ...
0
votes
2answers
25 views

Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$

This is actually part (c) of the original question. Part (a) asks to prove for any $h>0$, we have $\lim_{x\to\infty}\frac{f(x+h)-f(x)}{h}=b$. Part (b) asks to prove if $f(x) \to a$ as $x\to\infty$, ...
0
votes
3answers
64 views

Show that $(x^3+2x)/(2x+1)$ is $O(x^2)$

Show that $(x^3+2x)/(2x+1)$ is $O(x^2)$ The definition says: We say that $f(x)$ is $O (g(x))$ if there are constants $C$ and $k$ such that $$\mid f(x) \mid \leq C \mid g(x) \mid$$ whenever $x > ...
0
votes
0answers
15 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
-3
votes
2answers
48 views

Flourish in functions

Let $f$ be a continuous function defined on $[-2009,2009]$ such that $f(x)$ is irrational for each $x\in[-2009,2009]$ and $f(0)=2+3^{\frac{1}{2}}+5^{\frac{1}{2}}$. Find the value of $f(2009)$. I am ...
0
votes
1answer
26 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
1
vote
0answers
13 views

r(tau) within of H´s level curves

Im currently working on a math project and have one question regarding level curves. I´ve been given a function $$H(\theta,\omega)=\frac{1}{2}\omega^2-C\theta-\cos{\theta}$$ And I´ve shown that if ...
0
votes
1answer
41 views

Functions and range

$$ a \colon \mathbb{R}\setminus\{0\} \to \mathbb{R} \;\text{ defined by }\; a(x)= 6/x \\ b \colon \mathbb{Z} \to \mathbb{R} \;\text{ defined by }\; b(x) = 3x + 1 $$ a) State the range of ...
2
votes
1answer
21 views

Iterated limits of $\frac{x-y}{x^3-y}$

Why it the following limits look like this: $$\lim_{x\rightarrow -1} \frac{x-y}{x^3-y}=1$$ but suprisingly $$\lim_{y\rightarrow -1} \frac{x-y}{x^3-y}=\frac{1}{1-x+x^2}$$I thought that after ...
1
vote
1answer
27 views

Composite functions with domain and codomains

a: $ \mathbb{R} \to \mathbb{R}$ defined by $a(x) = (x/2) + 1$ b: $\mathbb{Z} \to \{0,1\}$ defined by $$b(x) = \begin{cases} 1 \qquad \text{ if } x \geq 1 \\ 0 \qquad \text{ if } x \leq 0 ...
1
vote
1answer
35 views

Function that determines angular velocity?

I see that someone posted the same problem a year ago, but the answer didn't quite give enough info. Here's the question: A movie crew is working on a scene that involves filming a car moving at a ...
0
votes
1answer
20 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
0
votes
0answers
13 views

respective values of two functions are “closer than expected”

let f,g be functions from the same finite set into the reals, let d be the mean distance between f(x) and g(x) for x in S, and let D be the mean distance between f(x) and g(y) for x,y in S; then D-d ...
3
votes
2answers
29 views

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies?

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies? I need help adding rigor to my observation to create a formal proof. ...
2
votes
5answers
54 views

Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable. Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Consider $g(x):= ...
1
vote
2answers
53 views

Can f have a finite limit at infinity?

The function $ f:\mathbb R\rightarrow \mathbb R$ is differentiable such that $f(0)=0$ and $(1+x^2)f'(x)\geq{1+(f(x))^2}$ for every $ x\in \mathbb R$ . Can $f$ have a finite limit at infinity?
0
votes
1answer
18 views

Number of Linear boolean-functions [on hold]

How many linear boolean functions are there, if we have n variable?
0
votes
1answer
14 views

$f:(2,4)→(1,3)$ where $f(x)=x-[x/2]$ (where $[.]$ is the greatest integer function/floor function),then what will be $f^{-1}(x)$.

Let $f:(2,4)→(1,3)$ where $f(x)=x-[x/2]$ (where $[.]$ is the greatest integer function/floor function),then what will be $f^{-1}(x)$? I can't understand how to manipulate the floor function.Help ...
0
votes
1answer
23 views

Logarith at base 10 as integration

The logarithmic function with base $e$ is the (set theoretic) inverse of exponential function $e\colon \mathbb{R}\rightarrow (0,\infty)$. This can also be defined using integration as: $\log\colon ...
1
vote
2answers
39 views

Power sets and functions

Let $a\colon\mathcal P(\mathbb N)\to\mathbb N$ be the function defined by $a(X)$ equals $0$ if $X$ has infinitely many elements and $a(X)$ equals the number of elements in $X$ if $X$ has finitely many ...
0
votes
0answers
14 views

How to find local extrema of f (p) give us area of triangle A1B1C1

For a right triangle ABC ( angle C = 90) on the rights height CC1 is chosen point P and consider the triangle A1B1C1 (A1 = AP cross BC, B1 = BP cross AC), if p is distance from point P to AB, to find ...
4
votes
3answers
46 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
1
vote
1answer
49 views

Is there always an upper limit for which $\int_0^l f(x)\,dx \; < \; \int_0^l xf(x)\,dx,$ is satisfied?

Given a function $f(x)$ which is strictly positive over all positive values of $x$ such that $f(0) = 0$, it makes sense to me by picturing what happens to $f(x)$ when you multiply it by $x$ that there ...
0
votes
3answers
60 views

List all functions f: {a, b, c} → {0,1 }.

This is a homework problem I have. Can someone just explains what it means, please? I can think of at least a dozen functions off the top of my head, but I think that's too many to be correct since we ...
2
votes
2answers
27 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
8
votes
2answers
11k views

Proving a function is onto and one to one

I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding. To prove if a function is one-to-one, it says that I have to ...
1
vote
4answers
47 views

Show $ex \leq e^x$ for all $x \in \mathbb{R}$

So far all I have is this: Let $f$ be a function where $f(x)=ex-e^x\leq 0$ $f'(x)=e-e^x \leq 0$, so $f$ is decreasing. I'm stuck here. Can someone help me with the next steps?
0
votes
0answers
18 views
+50

References for a notion related to radially lower semicontinuity

Let $E$ be a real vector space, $C\subset E$ be a nonempty convex set and $z\in C$. Let $f:C\rightarrow\mathbb{R}$ such that $$ \textbf{(A)} \quad f(z)\leq\limsup_{t\downarrow 0}f(z+t(w-z))\quad ...
0
votes
1answer
33 views

functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
1
vote
0answers
13 views

Verify combination of disjoint subsets $C$ and $D$ is onto

Let $C$ and $D$ be disjoint subsets of set $A$ and $f:C→B$ and $g:D→B$. Define a function $h(x)$ as follows: $$ h(x)=\left\{ \begin{array}{c} f(x) \textrm{ if } x∈C \\ g(x) \textrm{ if } x∈D ...
4
votes
0answers
28 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a ...
1
vote
2answers
63 views

Showing $(f^{-1}∘g^{-1})=(g∘f)^{-1}$

If the functions $f$ and $g$ are both bijections then the in inverse of the composition function $(f∘g)$ will exist. Show that it will be $(f^{-1}∘g^{-1})=(g∘f)^{-1}$ For the proof assume ...
1
vote
3answers
44 views

Parameterizing cliffs

I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to ...
-5
votes
1answer
35 views

Oil decay at 13%, how long until it is less than 21% of original?

My teacher gave me this problem, and it is very wordy, I don't really even understand what it is asking. First I took 100 and multiplied it by 0.13 subtracting that number from 100 and completing the ...
0
votes
1answer
48 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
-1
votes
0answers
75 views

What is the name of the function $f(x)=1/(1-x)$? [on hold]

I only want to know if the function $1/(1-x)$ has any specific name.
0
votes
2answers
37 views

Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
-1
votes
0answers
21 views

Continous frunctions problem

The problem says: f,g:[0;1]->[0,1] ,2 continous functions.They have the property that f(g(x))=g(f(x))). To solve: Both having the property of DARBOUX on the interval ,demonstrate that the numbers "c" ...
-2
votes
0answers
29 views

What does R[-a,a] represent?

More precisely: $f \in R[-a,a]$. All I could find was related to the symbol $\mathbb R$, but I have never seen it in this particular constellation, and even if it stood for "$\mathbb R$", I wouldn't ...
1
vote
1answer
72 views

Can a simple but rigorous argument be found to prove that this function is strictly increasing?

I have a problem here that asks to show that the function $ f: [0,\infty) \to \mathbb{R} $ defined by $$ f(x) \stackrel{\text{df}}{=} \begin{cases} \dfrac{1}{x} \left( 1 + \dfrac{x^{2}}{4} \right) ...
0
votes
2answers
25 views

Differential equations in function

Equations (1) : $xy'+(1-x)y=1$ let $z=xy+1$ determine and solve the differential equation (2) whose general solution is the function $z$ . -determine the general solution of (1)