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

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0
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2answers
60 views

Determinate a continuous function given a few coordinates.

Given a few Cartesian coordinate points $(x,y)$, for example: $(0, 1), (2, 3), (1, 4), (-1, -2),$ etc., is it possible determinate a function $y=f(x)$ that passes exactly for that points?
-1
votes
1answer
27 views

partially order

$( A,\le)$ and $(A',\le')$ are partially ordered sets. A map $\phi : A \to A'$ is called order preserving from $(A, \le)$ to $(A', \le')$ if for all $x, y \in A : x \le y \implies \phi(x) \le' ...
4
votes
5answers
318 views

Injection $\mathbb N \times \mathbb N \times \mathbb N \to \mathbb N$

I have to give an example of an injection $\mathbb N \times \mathbb N \times \mathbb N \to \mathbb N$. Would something like $f(x)=x^3$ be an answer to this question?
1
vote
2answers
241 views

if $f $ is continuous then is $\sqrt f $continuous?

I just want to know if $f $ is continuous on a compact interval, then does it follow that $\sqrt f$ is also continuous?
2
votes
1answer
34 views

Proving that $||f(x)||$ is Riemann integrable

Suppose $f=(f_1,...,f_m)$ be a vector valued function which is continuous on $[a,b]$. How can I show that $||f(x)||$ is also Riemann integrable? Any answers will be much appreciated. Thanks
0
votes
0answers
174 views

prooving the limit of a function in epsilon,delta, and sequences

Hi I need to prove the following limit of a function: I need to show the proof in two ways, one in epsilon and delta and the other with the sequence the diverge to infinity. I'm having difficulties ...
4
votes
1answer
142 views

How prove this $\displaystyle\lim_{\tau\to t}f(t,\tau)=\frac{1}{2\pi}\frac{x'_{1}(t)x''_{2}(t)-x'_{2}(t)x''_{1}(t)}{[x'_{1}(t)]^2+[x'_{2}(t)]^2}$

Question: let $$j_{1}(t)=\sum_{p=0}^{\infty}\dfrac{(-1)^p}{p!(1+p)!}\left(\dfrac{t}{2}\right)^{1+2p}$$ ...
3
votes
2answers
179 views

How to show the function function $f(a,b)=a+b\sqrt2$ from $\mathbb Q\times\mathbb Q$ to $\mathbb Q(\sqrt2)$ is bijection?

So I have this question that I've answered and don't know if it's correct: Show that $\mathbb{Q}(\sqrt{2})$ is denumerable. Proof: We define a function ...
0
votes
2answers
165 views

example for a strictly increasing function which has a discontinuous inverse

I want an example for a function $f$ which is strictly increasing in some subset $S$ of $ℝ$ such that $f^{-1}$ is not continuous on $f(S)$.I came up with my own example but I need to verify it. ...
0
votes
3answers
115 views

Does $f\circ h=g\circ h$ imply $f=g$?

Is this true or false? If $h\colon A\to B$, $g\colon B\to C$ and $f\colon B\to C$, and for all $x\in A$, $f(h(x))=g(h(x))$, then $f=g$. I seem to be able to prove it rather easily … or at ...
0
votes
1answer
27 views

Scaling fractions

I have a set of values between $0$ and $1$ inclusive and I want to scale them so that they are all between $0.5$ and $1$ inclusive. What function can I use to do this? \begin{align} f(0) &\to ...
0
votes
2answers
46 views

a misunderstanding about the definition of f(x)

well I have a little misunderstanding here if the functions is just a mapping tool then would not it make more sense to say that it can not take any value so to make my question a little clearer ...
2
votes
1answer
119 views

Verify if the function $\, f\colon \Bbb Z \to \Bbb R $ defined by $\, f(n)=n^3-3n$ is injective

I am stuck on the following problem : I have to verify whether the following two statements are true/false? The function $\, f\colon \Bbb Z \to \Bbb R $ defined by $\, f(n)=n^3-3n$ is ...
1
vote
1answer
32 views

Find the minium of function

Find the minimum of $f(x)=(x-1)(x-2)(x-3)(x-4)$ without using the calculus, I know it's easy to find it using the derivative, but I need to fiugre out how to solve it without it. I know that the ...
0
votes
2answers
39 views

Proving partition function equation

Let $\pi_m(n)$ represent the number of partitions of $n$ in which no part is greater than $m$. Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n).$ I know there is a theorem that helps, but I don't ...
0
votes
2answers
39 views

If $f$ is of given parity, what can be said of its derivative and primitive?

If $f$ is of given parity, what can be said of it's derivative and primitive? Clearly for power functions or simple trigonometric functions it seems that the parity of the derivative and ...
0
votes
1answer
629 views

Which function would best describe Moore's law

Moore's law states that the transistor density on integrated circuits doubles every 2 years. So this is an exponential function. My question is simple; what function of the form $y= a \times e^{bx+c}$ ...
4
votes
0answers
66 views

How prove this $f=C$ if $4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$

Question: if $f:\mathbb{Z}^2\to \mathbb{R}$ is bounded ,and for any $x,y\in \mathbb{Z}$,we have $$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$$ show that $$f\equiv C$$ where $C$ is constant. My ...
1
vote
1answer
54 views

When is $|f(x)|$ equivalent to $f(|x|)$

Specifically for functions of a complex variable. Are there any rules of thumb?
1
vote
2answers
2k views

Is $f(A)\cap f(B)$ a subset of $f(A\cap B)$?

Let $X$ and $Y$ be sets and let $f\colon X\to Y$ be a function from $X$ to $Y$. If $A$ and $B$ are subsets of $X$, is it true that $f(A)\cap f(B)$ is a subset of $f(A\cap B)$? If so, prove your ...
0
votes
2answers
253 views

Graphing letter A

How can I use two functions to graph the letter $A$? I need two functions $x(t)$ and $y(t)$ on $0\leq t\leq 1$ so that $(x(t),y(t))$ represents the alphabet$A$. I'm thinking of a $|x|$ and $x$. Can ...
2
votes
3answers
882 views

Intersection of normal to the curve

The line that is normal to the curve $x^2+3xy-4y^2=0$ at $(6,6)$ intersects the curve at what other point? If I implicitly differentiate this curve, I will get the equation of the slope: ...
1
vote
1answer
79 views

$\,f \colon \Bbb R^2 \to \Bbb R $ be continuous map such that $\,f(x)=0\,$ [duplicate]

I am stuck on the following problem : Let $\,f \colon \Bbb R^2 \to \Bbb R $ be continuous map such that $\,f(x)=0\,$ for only finitely many values of $x$. Then which of the following options is ...
0
votes
2answers
59 views

How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there …

I am stuck on the following problem when I was trying to solve an entrance exam paper: How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there with the property that $\, ...
5
votes
3answers
82 views

Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$

Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...
1
vote
0answers
53 views

A marginal density function problem

Given a plane with three points, $(0,−1), (2,0)$, and $(0,1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint continuous ...
-1
votes
2answers
36 views

Applying a function to a set rather than a value

I do apologize about the title, I dont understand the question so I couldnt come up with a better title, if someone else could edit it to a more meaningful title I would appreciate it. So here is the ...
0
votes
2answers
222 views

function composition - n times

Please consider this function: $$f(x) = \frac{x}{{\sqrt[6]{{1 + {x^6}}}}} $$ What would be the value of the composition (n times): $$f \circ f... \circ f = ? $$ I tried doing it manually, maybe ...
0
votes
1answer
47 views

Function composition and inverse

Consider f : ℝ \ {1} → ℝ \ {1} given by f(x) = x/(x-1) I need to find: 1) f ◦ f ◦ f and 2) the inverse function f^-1(x) So far I have: 1) f(f(x/(x-1)) = f(x) = x/(x-1) which is suspicious to me ...
1
vote
1answer
35 views

Solve for $x$ in $(3sf)$, where $\cos(x) - \tan(x) = 3 $.

The problem I am struggling with to solve is this. I have already tried to solve it however I end up with quadratic $\sin$ or $\tan$ whichever way you do it, which does not help. Solve for $x$ ...
2
votes
1answer
57 views

Show a function is one-to-one and onto

Consider f: ℝ{1} → ℝ{1} given by f(x) = x/(x-1) Show that f(x) is one-to-one and onto. What I have: If a function is one-to-one then it follows that if f(a) = f(b) then a=b. If a function is onto ...
2
votes
2answers
32 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
0
votes
1answer
68 views

Prove that the function is constant $f(x)=\arcsin2x\sqrt{1-x^{2}} - 2\arcsin x$ [closed]

Prove that this function is constant: $$f(x)=\arcsin\left(2x\sqrt{1-x^{2}}\right) - 2\arcsin x$$
0
votes
2answers
83 views

$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded? [duplicate]

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded? I think that it will always ...
1
vote
1answer
32 views

Prove Bijection in roots of unity function

Given $k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} $. Probe that if $n$ and $m$ are coprime, the function $f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta$ is bijective. ...
0
votes
1answer
46 views

Boolean function on $\{0,1\}^n$ comprising just binary AND and OR gates

Let $f:\{0,1\}^n\to\{0,1\}$ be a boolean function computed by logical circuit comprising just binary AND and binary OR gates (assume that the circuit doesn't have any feedback). Let ...
5
votes
1answer
68 views

function inequality $f(x+y)+y \leq f(f(f(x)))$

$f(x+y)+y \leq f(f(f(x)))$ find all possible solution for $ f: \mathbb {R} \rightarrow \mathbb {R}$
0
votes
2answers
442 views

How to show this function is an injection (one to one)?

Consider the function $f: \mathbb N$ × $\mathbb N$ → $\mathbb R$, $f(a,b) = a+b \sqrt{11}$ How do I show this function is an injection (one to one)?
2
votes
1answer
25 views

Extending $\mathbb R$ for the benefit of the unitary pulse function

The unitary pulse function (or sample function) is defined as follow: Let's $\newcommand\R{\mathbb R}d_1:\R\to\R$ be a positive intrgrable function such that $$\int_{-\infty}^{\infty}d_1(x)dx=1.$$ ...
0
votes
2answers
57 views

If $ 2<x^2<3$ then find then no of solutions which satisfy that $({x^2})=1/(x)$

If $$ 2<x^2<3$$ then find then no of solutions which satisfy that $$({x^2})=1/(x)$$ where (x) stands for fractional part.! I tried to convert it into greatest integer but i got no ...
0
votes
1answer
70 views

Prove that [x/y] is a primitive recursive function

Prove that [x/y] is a primitive recursive function using this theorem: If $g(x_1,...,x_n)$ is primitive recursive, then $f(x_1,...,x_n)=\sum^{x_n}_{i=0}g(x_1,...,x_{n-1},i)$ is also a primitive ...
1
vote
0answers
169 views

composite functions and injection

I want to know if my attempts at parts b,c are correct for the following problem. a)Prove that the composition of one to one functions is a one to one function. b)Show by example that the converse is ...
0
votes
1answer
88 views

computability and uncomputability

1) Suppose $f$ is an increasing function from $\mathbb N \to \mathbb N$ $(i.e., if x\ge y, then \space f(x) \ge f(y)).$ Is there necessarily a program which computes $f$? 2) Suppose $f$ is a ...
1
vote
2answers
64 views

How to proof the randomness of a number sequence?

I've got a sequence of numbers generator by a "random number generator". Is there a way or a method to proof the randomness of the generator? How would I even compare randomness of generators? Or ...
6
votes
2answers
113 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
0
votes
4answers
63 views

Connected Subsets of X x Y

Let $X$ be a connected topological space and $f : X\to Y$ a map. Show that the graph $G(f)$, defined by $G(f) = \{(x; f(x)) \in X \times Y | x \in X\}$ is a connected subset of $X \times Y$. I ...
2
votes
1answer
57 views

Does there exist a suitable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n^m) = mf(n)$? I think no.

Any ideas how to prove that no injection $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is closed under multiplication by elements of $\mathbb{N}$ satisfies the following identity? $$f(n^m) = ...
1
vote
1answer
26 views

Number of distinct functions on a vector space $\mathbb{N}^3$

Let $k$ be an integer at least $4$ and let $[k] = \{1,2,\ldots,k\}$. Let $f:[k]^4 \to\{0,1\}$ be defined as follows: $$f(y_1,y_2,y_3,y_4) = 1\ \mathrm{iff\ the\ y_i's\ are\ all\ distinct}$$Now for ...
0
votes
2answers
38 views

Function that is onto and continous

Can there be a continuous onto function f : R -> R \ Q? I know if a function existed it would map reals to the irrationals. Any ideas?
2
votes
2answers
193 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...