Tagged Questions

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

learn more… | top users | synonyms (1)

0
votes
0answers
12 views

Bound on functions part 2

I'm trying to replicate the following idea in a slightly different setting, Suppose we want the set of $r$ such that the condition $(*)$ $|I_{r}| \leq O(r^{2})$ and $|I_{r}| \geq \rho^{r}$ where ...
3
votes
0answers
59 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function. Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. Let $k\in\mathbb{N}^*$. What can be said of the ...
0
votes
1answer
11 views

Bound on function

Suppose a set $|I_{r}| \leq O(r^{2})$ but also $|I_{r}| \geq \rho^{r}$ where $\rho=(1+\epsilon).$ Why is it true that there must always be an $r^{*}=f(\rho)$ such that the condition is true for all ...
-2
votes
0answers
19 views

Extend $f: [-1,1]\to\mathbb R; x\mapsto |x|$ regularly to $\mathbb R$. [on hold]

Can you answer this exercise? Consider $f(x)=|x|$ with $x\in [-1,1]$ and extend it regularly to $\mathbb R$. Give a rigorous expression of this function. Thank you.
1
vote
1answer
25 views

How to tell whether or a function is surjective or injective?

If one is given the following: $$A = \{(x, y)\mid x \in \mathbb{R}, y \in \mathbb{Z}, y = \lceil x \rceil\},$$ a relation from $\mathbb{R}$ to $\mathbb{Z}$. How would I be able to tell whether ...
0
votes
2answers
30 views

Functions from a set of numbers to a set of letters?

Say I have two sets, $A = \{1, 2, 3\}$ and $B = \{a, b, c\}$. I know how to find regular functions with all numbers however how do I find a function that is $f: A\to B$?
0
votes
1answer
19 views

Is this function one-to-one and onto?

Let $$f : \mathbb{R} → \mathbb{R}, f(x) = 3^3 + 2$$ I know it's not onto actually, because it doesn't give all the real numbers. But is it one-to-one, even though we're not actually using the x ...
-2
votes
1answer
12 views

Limits-related Question (2) [on hold]

Say I have two functions $$f,g: (k,\infty) \to R_+$$ where $k>0$ and $ f(x)<g(x)$. Under what circumstances can I deduce the following? $$\lim_{x\to \infty}f(x)<\lim_{x\to \infty}g(x)$$ ...
0
votes
0answers
10 views

Images of some regions of the complex plane by given function?

I'm trying the draw the image of $A=\lbrace z\in \mathbb{C}:-1<Im((1+i)z)<1\rbrace$ by $f(z)=1/z$ and the one of $B=\lbrace z\in\mathbb{C}:|z|<1\rbrace$ by $f(z)=(z-1)^{-1}$. I've managed to ...
1
vote
1answer
21 views

Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not true that $E\left(e^X\right)=e^{E(X)}$. However, when one conditions on $X$, is this true? Does it ...
6
votes
1answer
46 views

A function is smooth at a point and not smooth in any neighbourhood of it, exist or not?

Suppose that a function $f$ defined in an open set $U \subseteq \mathbb{R}^m$ is smooth at a point $p \in U$. Then we have that there exists an open set $U_n \subseteq U$ $($ say $U_{n+1} \subseteq ...
1
vote
1answer
16 views

How to calculate amount of boxes needed for X amount of items

I'm looking for a function that can give me the amount of boxes needed for a given amount of items. And if possible, that it gives me an equal, or close to equal, distribution of items in each box. ...
1
vote
0answers
28 views

Does a specific function exist with these properties?

lets say i have a function where: $[p,c,n,k] \in \mathbb{Z}$ defined some way in this function $ f(x) = \sum_{q=0}^{n} \sum_{v=1}^{k-q+1}\frac{c(k+1-v)!(p-k)!}{(k+1-v-q)!q!(p-k-q)!}x^{k-v+1-q} = ...
-3
votes
0answers
20 views

Proof by contradiction in continuity

Let $f$ and $g$ be functions from $[0;1]$ to $[0;1]$ and continuous on $[0;1]$ So that for all $x \in [0;1]$ we have $(f\circ g)(x) = (g\circ f)(x)$. Question : Show using proof by contradiction that ...
0
votes
0answers
19 views

Function which has the following properties [on hold]

I wanted a function which is strongly positive (sharp gradient) in the ++ and positive in the +, and the same in minus
0
votes
0answers
12 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finit, with ...
1
vote
1answer
29 views

An Injective binary function

So I work with theoretical chemistry and time and again we end up walking into what is, for us, uncharted mathematical territories in our search for the patterns and symmetries of nature and its ...
0
votes
1answer
17 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
2
votes
2answers
25 views

Inverse of a function - $f^-(x)$ in a restricted domain

I have a function $f(x) = y = x^2(3-2x)$, $0\le x\le 1$, I would like to find the inverse of this function on this restricted domain. ,This seems to be the answer for one of the problems that I have ...
0
votes
1answer
21 views

Sequential criterion for functional limits proof in the opposite direction

Let $f: A\to\mathbb{R}$ Given $c$ is a cluster point in $A$. Prove that the following statements are equivalent: (a) The function $f$ does not have a limit at $c$. (b) There exists a sequence ...
2
votes
1answer
17 views

Write an equation as a single power(Grade 11 Math, Function)

10 to the power -4/5(10 to the power 1/15) divided by 10 to the power 2/3 The answer is 10 to the power -7/5 which seems impossible to me. I get: 10 to the power -4/5(10 to the power -11/15) I see ...
4
votes
3answers
48 views

Limit theorems, prove function has a limit at every point

Suppose that $f:R\to R$ is a function such that $f(x+y)=f(x)+f(y)$ for all $x,y∈R$. Assume that $f$ has a limit at $0$, $f(1)=1$. Prove that $f(x)=x$ for all $x \in R$ Hint: Show first that $f$ is ...
0
votes
1answer
33 views

Well defined Functions on Congruence classes

Could someone please confirm my logic or point me in the right direction? Thank you. 1) Is the function $f : [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = [n^2]_p$ well defined? 2) Is the ...
0
votes
2answers
14 views

Questions about the addtion of injective and surjective functions

I have a question which is as follows, Consider $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$. Define the sum of $f$ and $g$ as the funtion $h:\mathbb{R} \to \mathbb{R}$ such that ...
0
votes
0answers
24 views

Function Couting

I Have question, maybe You Can help :) (sorry, im don't understand method chain) X = {1, 2, 3, {1, 2}}; Y = {1, 2, a, b, c}. How much is the all functions f : X -> Y? How much is Injective ...
1
vote
1answer
21 views

Joint Probability Function

Two hats are drawn randomly w/o replacement from box containing $8$ black, $4$ red, and $2$ yellow hats. If $X$ denotes the number of black hats drawn and $Y$ the number of red hats drawn. What is the ...
0
votes
1answer
12 views

Injective, Surjective functions help

f : N → N I am seriously struggling with finding examples of functions f : N → N for: f is neither injective nor surjective, f is injective but not surjective, f is surjective but not injective, f is ...
0
votes
1answer
16 views

Function distributing over intersection of sets

Let $\alpha : S \to T$ be one to one, and let $A$ and $B$ be subsets of $S$. Assume that $S$, $T$, $A$, and $B$ are nonempty. Show that $\alpha(A\cap B) = \alpha(A) \cap \alpha(B)$
0
votes
0answers
41 views

Creating two functions $ h ( x ) $ and $ g ( x ) $ such that $ f ( i , j ) = f ( h ( x ) , g ( x ) ) $

So as mentioned in the title I want to create two functions $ h ( x ) $ and $ g ( x ) $ such that $ f ( i , j ) = f ( h ( x ) , g ( x ) ) $ For simplicity I will refer to $ f ( h ( x ) , g ( x ) ) $ ...
0
votes
3answers
14 views

Slope from a graph

I was wondering if I have a graph with no points specified and they only say find the slope how do I do it like what am I supposed to do and how can I find out the points
2
votes
3answers
107 views

How to prove properly that $\mathbb{N \times N}$ $\rightarrow$ $\mathbb{N}$ : $(p,q) \rightarrow \frac{(p+q)(p+q+1)}{2} +q$ is a bijection?

I tried to show that for : $\frac{(p_1+q_1)(p_1+q_1+1)}{2} +q_1$=$\frac{(p_2+q_2)(p_2+q_2+1)}{2} +q_2$ we have $(p_1,q_1)=(p_2,q_2)$ to prove that it's an injection. But I obtain : ...
0
votes
4answers
34 views

Is there a continuous function f(x) that tends to zero for both positive and negative infinity and f(0) = c

Is there a continuous function f(x) that tends to 0 as x approaches both positive and negative infinity and f(0) = c, where c is some given constant?
0
votes
0answers
35 views

Well-defined functions using mod $p$ equivalence classes

Prove that if $m, n$ are elements in the set $\mathbb{Z}$ and $m \equiv n \pmod p$, then $m^2 \equiv n^2 \pmod p$. Also, is the function $f: [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = ...
1
vote
2answers
27 views

Well-defined functions on equivalence classes

Could someone please explain how to develop a proof and the reasoning behind the following: My professor said that to determine if a function is well-defined, we check to see if equivalent elements ...
1
vote
0answers
14 views

Comparing function to parent function without graphing

How can I compare this function to the parent function without graphing? Where did the 5/4 come from and what steps do I need to take to solve this?
0
votes
1answer
22 views

Finding constants to make f(g(x)) and g(f(x)) equal.

Let $f(x) = ax + b$ and $g(x) = cx^2$, where $a$, $b$, and $c$ are constants. Compute $f\circ g$ and $g\circ f$. Determine for which constants $a$, $b$, and $c$ it is true that $f\circ g =g\circ f$. ...
0
votes
1answer
25 views

Equivalence Relation and functions question

Could someone please confirm if I understand this correctly? Here is the problem: define ~ on Z by m ~ n in case m^2 ~ n^2. 1) What, if anything, is wrong with the following "definition" of a ...
3
votes
3answers
34 views

Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
0
votes
0answers
16 views

What happens when log domain changes

For example, let's say we have (log is defined base e) $ \log(\log (n^2))=\log(2\log n)=\log2+\log(\log n) $ When applying the rules, we clearly see that the domain of n changed. My question is that ...
0
votes
1answer
25 views

Does it exist a function that is continuous at every rational point and discontinuous at every irrational point?And vice versa?

Actually there are 2 questions, but they are closely related. Does it exist a function that is: 1. Continuous at every rational point and discontinuous at every irrational point? 2. Continuous at ...
-2
votes
0answers
43 views

Mathematical Language Functions [on hold]

Let $S$ be the subset of $\mathbb{R}^2$ defined by $$S=\{(x,y)\in\mathbb{R}^2:−3≤x≤−1, 0≤y≤1\},$$ and let $f$ and $g$ be the functions defined by $$\begin{array} {rcl}f : & \mathbb{R}^2 & ...
0
votes
2answers
15 views

Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism.

Let V, W be finite dimensional vector spaces. Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism. Thanks in advance! I ...
0
votes
0answers
25 views

About a function with degenerate non-isolated critical point

A few books use function $f(x)=e^{-1/x^2}\sin^2(1/x)$ as an example with 0 as its non-isolated degenerated critical point. And it's graph is showed as Is this ...
0
votes
2answers
20 views

Prove / Disprove function

Let $f(n)$ and $g(n)$ be arbitrary functions from $\mathbb{N}$ to $\mathbb{R}^{+}$. Prove or disprove the following: $$f(n)+g(n) = \Theta(\min\left \{ f(n), g(n) \right \})$$ Please help me prove (or ...
2
votes
1answer
34 views

Functional-equation for strictly increasing functions

Let $n,c$-given natural numbers .Let $f(x)$ - strictly increasing function , domain of definition and set of values of ​​which are non-negative integers ,$f(0)=0 , f(1)=c$ and \begin{align} ...
0
votes
2answers
27 views

Simplest Forms of a function

If $f(x)=2x-3$ and $g(x)=x^2 + 2$ how do i find the simplest forms of: $f(g(x))$ and $g(f(x))$ Is it just a matter of substituting the equation in or?
4
votes
1answer
50 views

Find $f$ and a $g$ function given that

Find an $f$ and a $g$ function given that $$f(g(x)) = \sqrt{1-x^2},\\ g(f(x)) = \left(\frac{x-2}{x+1}\right)^2$$ I'm a bit confused on this one. Would $g(x)$ and $f(x)$ for the two equations be ...
0
votes
1answer
39 views

Find a map $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ to prove surjectivity for a given $f:\mathbb{R} \rightarrow \mathbb{R}^2 $

When the following is given: Let $f:\mathbb{R} \rightarrow \mathbb{R}^2 $ be given by $f(x)=(4x, -x)$ for all $x \in \mathbb{R}$ How to find a map $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
1
vote
2answers
15 views

Function Notations for quadratics

Given $T(x) = ax^2 + bx + c$ Find $a,\,b$ and $c$ if $T(0) = -4$, $T(1) = -2$ and $T(2) = 6$ I first made $C$ for when $x = 0,\,-4$. But don't know were to go from here. Any help?
3
votes
2answers
194 views

What method can i use to find the first 3 roots of y(t)=tan(t)+t?

Just by looking at the function: $$y(t) = \tan(t)+t$$ I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer ...