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

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1
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1answer
25 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
0
votes
1answer
18 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
0
votes
1answer
32 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
-2
votes
0answers
13 views

how to increace the volume to a specific volume in revolution of solid, using integration [on hold]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
2
votes
3answers
44 views

Show that $f(x)=x/\sqrt{x^2+1}$ is a bijection of $\mathbb R$ onto $\{ y: -1<y<1\}$

I am looking for help in regard to a practice question about functions. The question is Show that a function $f$, defined by $f(x)=x/\sqrt{x^2+1}$ , $x \in \Bbb R$ is a bijection of $\Bbb R$ onto ...
0
votes
2answers
31 views

If a mapping and it's inverse are both one to one, then must the mapping be bijective?

If $\sigma$: $A$ $\rightarrow$ $B$ was a mapping which was one to one, and had an inverse $\sigma$$^{-1}$: $B$ $\rightarrow$ $A$ which is also one to one, then are they both bijective mappings? I'm ...
0
votes
1answer
40 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
2
votes
1answer
36 views

Confusing about the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$

What is the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$? A nice plot of $f(x)$ shows that the domain is $\mathbb{R}$ but we see that $x$ should be non-negative at the first sight. Of course, I ...
0
votes
1answer
37 views

How to prove a function from a set of triples to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection?

Let $Y=\{y_1, y_2, y_3, y_4,y_5\}$ The function from the set of triples $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$ to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection given by ...
1
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3answers
49 views

Should real functions be described as $ f: \mathbb{R} \rightarrow \mathbb{R} $ or $ f: \mathbb{R} \rightarrow \mathbb{R}^2 $?

I've been trying to teach myself topology, and I'm having a bit of trouble grasping the abstract concepts of the field. One question that's been poking at my understanding regarding topological ...
1
vote
2answers
62 views

How to find the domain and range of $f(x) = \sqrt{x^2-2x+5}$?

This is the function: $$f(x) = \sqrt{x^2-2x+5}$$ Edit: normally what I would do is this: Since it's a square root function, the thing inside the root has to be $\ge 0$. So, $(x^2 - 2x+5)\ge 0$. Then ...
0
votes
1answer
23 views

Why does this combination correspond to an injection from $\mathbb{N_2} \rightarrow Y$?

Suppose 3 people each select a main dish from a menu of five items. How many distinct choices are possible if 2 people select the same dish? The solution: Let $X$ be the set of 3 people and $Y$ be ...
1
vote
2answers
25 views

Composite Relations

I'm new to functions and relations, and I've only just figured out that there are 16 relations on a set with 2 elements. I can't figure out what is meant by R ; R ⊆ R other than the fact it is a ...
2
votes
1answer
37 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
0
votes
2answers
36 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
-2
votes
0answers
18 views

Monotonicity of derivative [on hold]

Suppose $f$ is a twice continuously differentiable function with linear growth. Prove that if there exists a $\delta$ such that for $x>\delta$, $f/x$ is monotonic, then there exists a $\delta_2$ ...
0
votes
2answers
60 views

$\lim_{x \to \infty} f(x)=1 $ $\implies$ $f(x) \sin x$ is uniformly continuous on $\mathbb R$?

Let $f:\mathbb R \to \mathbb R$ be a continuos function such that $\lim_{x \to \infty} f(x)=1 $ , then is it true that $f(x) \sin x$ is uniformly continuous on $\mathbb R$ ?
0
votes
1answer
39 views

Intersection of Images of a function

I'm trying to understand intuitively why the image ( under some function ) of the intersection of subsets of the domain of that function is only contained ( and not equal ) to the intersection of the ...
9
votes
3answers
465 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
2
votes
1answer
18 views

How is it called when you apply min / max seperatly to each dimension?

I want to do the following: $$\begin{pmatrix}3\\1\\4\\1\end{pmatrix} = \min( \begin{pmatrix}4\\4\\4\\4\end{pmatrix}, \begin{pmatrix}3\\1\\4\\10000\end{pmatrix}, ...
2
votes
1answer
79 views

How to find all complex polynomial $f$ such that $1+f(z^n+1)=(f(z))^n$

Question: Let $n\gt1$ be a natural number. Is there a non-constant complex polynomial $P$ such that $P(x^n+1)=P(x)^n-1$ for all $x$? I saw this problem about polynomial, here is the question: Find ...
3
votes
2answers
59 views

Intuitive and convincing argument that functions are vectors

Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the ...
4
votes
2answers
34 views

What is a transformation?

I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld ...
0
votes
0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
2
votes
1answer
53 views

Method for computing limit of a sin function as x tends to zero

I have a question about computing $$ \lim_{x \to 0} \sin\left(\frac{\pi x}{4|x|}\right)$$ I found the limit of $\pi x$ and $4|x|$ seperately and ended with $\sin(\pi/4)$ which is equal to ...
0
votes
3answers
56 views

Method for computing limit of a function as $x$ tends to zero

I have a question about computing $$\lim_ {x \to 0} \dfrac{(2/x^3)+(1/x^2)+(1/x)+1}{(1/x^3)+1}.$$ I used a shortcut method of dividing by the highest power but I don't think that I can use this method ...
0
votes
0answers
48 views

Elementary Pigeonhole Principle Question

Is my reasoning here correct? If not, advice would be appreciated. Thank you for your time! We assume that $A$ is finite and $f: A \rightarrow A$. We show that $f$ is one-to-one iff $ran \ f = A$. ...
2
votes
0answers
37 views

notation for minimum and maximum?

I'm trying to figure out the correct notation for this situation for use in Machine Learning. I have various ratings (for texts): ...
1
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2answers
85 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
0
votes
1answer
22 views

Objective function with two variables

A factory produces jointly two articles, and it has the problem to decide their prices in order to maximize the monthly income, knowing that the demand d1 (in hundreds of units) of the first article ...
7
votes
5answers
480 views

Functional Notation.

I have some doubts regarding function notation: First If I present a function I write:$f(x)$ If I write it's inverse:$f^{-1}(x)$ So why doesn't$f(f(x))=f^2(x)$ Second If $\frac{df(x)}{dx}=f'(x)$ ...
0
votes
1answer
50 views

Limit of a function is unique [on hold]

I have read the proof of this property. The uniqueness of a limit of a function: Spivak's proof I was thinking we can also prove this informally by using the definition of function. For a ...
2
votes
2answers
63 views

Proving that a function satisfying $|f(x)-f(y)| \leq |x-y|^3$ is constant [duplicate]

Let $\mathbb R$ be the set of real numbers and $f: \mathbb R \rightarrow \mathbb R$ be such that for all $x$ and $y$ in $\mathbb{R}$, $$|f(x)-f(y) |\leq |x-y|^3.$$ Prove that $f(x)$ is a constant. ...
5
votes
2answers
76 views

What is this notation for a function? I've never seen it written like this before.

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$ I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside? ...
1
vote
4answers
39 views

How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
1
vote
0answers
54 views

Is $f^n_b$ a surjection?

For the purposes of this question, let $\mathbb{N}_k = \mathbb{N}_{\geq k} = \left\{n \in \mathbb{Z} | n \geq k \right\}$ and $\mathbb{R}_+ = \mathbb{R}_{\geq 0} = \left\{x \in \mathbb{R} | x \geq 0 ...
2
votes
1answer
27 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
0
votes
2answers
46 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
-1
votes
1answer
33 views

Typical objective question on functions [closed]

Let $f$ be a real valued continuous function on $[0, 3]$. Suppose that $f(x)$ takes only rational values and $f(1) = 1$. Then $f(2)$ equals:$\begin{array}{l} \mbox { (a) } 2 \\ \mbox { (b) } 4 ...
4
votes
1answer
66 views

Is there a name for the property of a function f such that $f(x,y)=f(y,x)$?

As in the title: is there a name for the property of a function such that $f(x,y)=f(y,x)$. I don't know how to be clearer than that. I tried to look for symmetric property on Google, but without any ...
1
vote
1answer
54 views

Implementing a function in PARI/GP

I want to define a function: $$g(n)= \begin{cases} +1 & \text{if $n=1$},\\ +1 & \text{if $n$ is an odd indexed prime}, \\ -1 & \text{if $n$ is an even indexed prime},\\ (-1)^r & ...
2
votes
0answers
42 views

What is the curve's name for the “reciprocal” equation of a circle?

The equation of a unit circle is $$(x-a)^2+(y-b)^2=r^2$$ When the origin $$(a, b)=(0,0)$$ the equation becomes $$y=(1-x^2)^{1/2}$$ Naturally when this equation is plotted on graph paper we get a ...
0
votes
2answers
48 views

$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
0
votes
2answers
15 views

Simplify and Evaluate function of X

I am having problems understanding the answer to this question Let $f(x) = 1 - x + 4x^2 $ Evaluate $(f(x+h)-f(x))/h$ Any help would be much appreciated, Thanks
1
vote
1answer
19 views

Intervals on which function is increasing and decreasing

Let $p(x)=x^5-q^2x-q$ , where $q$ is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below. We compute ...
0
votes
1answer
20 views

Blend n number of values by distance

I have n number of values which each have a distance that determens how much of the amount that should be blended. I've tried to illustrate my problem visually: The blue numbers is the values, the ...
20
votes
6answers
2k views

Is there a name for the function max(x, 0)?

Is there a name for the function $ \max(x, 0) $? For comparison, the function $ \max(x, -x) $ is known as the absolute value or modulus of x, and has its own notation $ |x| $
2
votes
1answer
38 views

$\tan(x), \cot(x)$ function properties

Does $\tan(x)$ and $\cot(x)$ has symmetry axis? (like e.g $\cos(x)$ at $\pi k$ for $k \in \mathbb{Z}$), I tried think in the direction that $\sin(x)/\cos(x) = \tan(x)$ and both of them have symmetry ...
1
vote
1answer
30 views

fundamental period of sum of two periodic functions

Is there some formula to find fundamental period of sum of two periodic functions both of whose fundamental period is known. If yes what is the proof and the formula
1
vote
3answers
76 views

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's ...