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

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2
votes
1answer
83 views

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto?

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto? I am not sure how to tell. Say $b\in N\times N$ this means the codomain is all the different combinations of the natural numbers. But ...
2
votes
2answers
24 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
3
votes
3answers
114 views

Finding if a function is onto?

Is the following function onto? It is a piece-wise function. Let the function $f:\mathbb{R}\rightarrow \mathbb{R}$ be $f(x)= \begin{cases} 2-x &, x\le 1 \\ \frac{1}{x} &, x>1 ...
2
votes
1answer
39 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
1
vote
1answer
33 views

Composition of functions which is one-to-one.

$f:Y\rightarrow Z$ and $g:X\rightarrow Y$ If $f\circ g$ is one-to-one then which of the following must be true? 1.$g\circ f$ is one-to-one. 2.g is one-to-one. 3.f is one-to-one. 4.g is onto.
1
vote
1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
0
votes
2answers
69 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
0
votes
0answers
39 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
2
votes
3answers
141 views

How to find know if function is onto?

How do you figue out whether this function is onto? $\mathbb{Z}_3\rightarrow \mathbb{Z}_6:f(x)=2x$ Onto is of course is for all the element b in the codomain there exist an element a in the domain ...
1
vote
3answers
105 views

Is $4x^2-4x+2$ surjective?

Determine whether the function $f_4:\mathbb{R^+}\rightarrow \{x \in \mathbb{R^+} x \ge 1\}$ given by $f_4(x)=4x^2-4x+2$ is injective, surjective or bijective. I will just show parts of the solution I ...
0
votes
0answers
14 views

Functions to fill spheres at certain points in a cylindrical volume?

I know how to find how many spheres of a given radius can fit into a cylinder. But, I also want 2 know how to fill them throughout points in the cylinder. To give an example I have spheres of a ...
1
vote
3answers
107 views

Why isn't an injection an iif?

Suppose that $f:X \rightarrow Y$ is a function. Then an injection can be defined as: $\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$ Why isn't it defined instead as follows: $\forall ...
0
votes
0answers
28 views

Reversible smoothing of a two dimensional function (or an image)

Smoothing of an image, or a two dimensional function is quite easy, there are many methods to achieve it, using average of near elements. But how to make it reversible? Maybe DCT (discrete cosine ...
-1
votes
1answer
18 views

Rewriting a quadratic function

i have to find domain of this function $f(x) = \log(10+3x-x^2)$ can i rewrite this as $\log(x^2-3x-10)$? I found the domain but is not the same for 2 forms of function.
1
vote
2answers
51 views

Problem Involving a Generalized Cartesian Product

Let $I$ be a set, and for each $i \in I$, let $U_i$ and $V_i$ be sets. Furthermore, suppose for each $i \in I$, there is a bijection $f_i:U_i \to V_i$. Prove that there is a bijection $g:\prod_{i \in ...
0
votes
1answer
29 views

Need function for 2D sigmoid-shaped monotonic Surface

I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ ...
0
votes
3answers
66 views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
0
votes
4answers
35 views

Showing that $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f(x)=[3x]$ is function?

How can I show this is a function? $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f([x]_4)=[3x]_2$ where $[x]_n$ is the equivalence class of $x\mod n$. I think it is a function because I cannot ...
3
votes
2answers
54 views

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$.

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. EDIT: Actually, this identity should hold even if $f$ is not one-to-one (injective), right? ...
6
votes
1answer
69 views

What am I doing wrong in this algebra excercise?

This is my first question here, so please forgive me if the format etc. are not quite right. I've been attacking an algebra question, and my workings are below. There's a mistake somewhere (I don't ...
1
vote
3answers
181 views

How find this P(x) if $ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) $

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that $$ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) $$ This problem is IMO Shortlist ...
0
votes
1answer
42 views

Derivation of Dirac-Delta with complicated argument $\delta(f(x))$

Recently I learned how to deal with the Derivative of a shifted Dirac-delta. Now I want to go a step further, but are not sure about the solution. Is there a simple way to rewrite terms like this ...
1
vote
3answers
56 views

True or False Question About Functions [closed]

If $f(1)>0$ and $f(3)<0$, then there exists a number $c$ between $1$ and $3$ such that $f(c)=0$. I'm not sure how to solve this question. Thanks in advanced!
0
votes
1answer
30 views

What is the domain? [closed]

What is the domain of the function $f(x) = \sqrt{4 x + 37}$? I am not sure where to get started with this problem. Thanks in advance!
0
votes
0answers
30 views

Find domain of a function

Function is : $f(x) =\sqrt{ 16 - x^2}$ First i change side $x^2-16\ge0$, then $(x+4)(x-4)\ge 0\implies x=-4,x=4$. I found that domain is:$(-\infty,-4] \cup [4,+\infty)$ but the problem is: this ...
-2
votes
4answers
53 views

Solving a problem using the definition of limit [closed]

How can I solve this using the definition of limit? Prove using the definition of limit that: $$\lim_{x\to 1} (x²-4x)=-3$$ How can I approach this? EDIT: OH my god! Thanks @adam! Maybe you ...
0
votes
0answers
30 views

Sketching the graph of a function with three real roots

I need to solve the following question: Sketch a graph of a function $f(x)$, continuous in all $x \in \Bbb R$, knowing that $f$ has three real roots, that $\lim_{x\to+\infty} \left[f(x)-\frac ...
2
votes
3answers
38 views

Is the inverse of this function unique

Let $f$ be a function from any set(Say $K$) to any set (say $P$) Now: $f(x)=2x+1$ My question:Is it necessary that the inverse of the function is $\frac{x-1}{2}$? This is a problem given in my ...
0
votes
1answer
31 views

Constructing the graph of a function

I need to solve the following problem: Consider the function $g(x) = \ln(x^{2}) + 2$. Construct the functions graph $f(x)=\int g(x)\:\mathrm{d}x$ considering the integration constant equal to ...
0
votes
0answers
18 views

Property of a function.

If $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ s.t $f(1,n)=n$ and $f(m,n)\geq f(m-1,n^2+n)$. For such a function,is the following true $x \geq y \implies f(m,x) \geq f(m,y) \ \forall m\in ...
1
vote
2answers
20 views

Prove that addition of a constant on vector spaces is bijective

What would be a nice way to deduce from the vector space axioms that $f : V_1 \longrightarrow V_2, \, x\mapsto x+v$ with constant $v$ is bijective?
0
votes
1answer
31 views

Surjectivity of composition

I know that this question has been posted few times, but I want to check MY proof, because this is my first time trying to prove anything in mathematics. (I'm afraid if I just copy paste their proofs ...
0
votes
1answer
41 views

Understanding the difference between relations and functions.

$R=\{(1,2),(1,3)\}$ is a relation but not function. The logic for this is that if the first element of every ordered pair must remain different, then it is said to be function. Otherwise, it's just ...
-3
votes
1answer
54 views

A function $g(x)=ax^2 + b/x$ has a minimum value at $x=4$. [closed]

A function $g(x)=ax^2 + b/x$ has a minimum value at $x=4$. The function value at $x=4$ is $96$. Determine the value of $a$ and $b$.
0
votes
1answer
16 views

Question about strongly convexity and affinity

For a function $f$, it is said to be strongly convex if for all $x,y$ \begin{equation} (\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 \end{equation} for a constant $m \ge 0$ Is it called ...
1
vote
3answers
204 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
0
votes
0answers
5 views

tangent vector on cone

Suppose that $C$ is a cone with vertx at the origin and let $\nu(x)= (\nu_1(x),\cdots, \nu_n(x))$ is the tangent vector at the point $x \in \partial E$. Is $\nu(x)$, (as function of $ x \in \partial ...
0
votes
1answer
25 views

Logarithm with variable base

I am trying to define a function that maps polynomials in the form of $x^{3^n}$ to the value of $n$ in the polynomial, where $n\in{Z}$.* Is is valid to define this function as $log_{x^3}(u)$, where ...
3
votes
2answers
59 views

Function for this game movement graph?

This may be too easy for you, but here goes: I'm creating a kind of ski slalom game where I want the horizontal speed/direction to follow the attached graph. X is time, Y is horizontal speed. Positive ...
1
vote
3answers
58 views

Example of a bijection from the set of real numbers to a subset of irrationals

I need an example of a bijection from the set of real numbers to a subset of the irrationals. I tried something like $f(x)=x+\sqrt{2}$, but where should I map $-\sqrt{2}$?
0
votes
3answers
43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
0
votes
1answer
22 views

Analyzing a particular type of functions

Let $f$ be a function from $\mathbb{Z}$ to $ \mathbb{Z}$ Now $f(x)=x$ Question: Is $f$ continuous in its domain?( perhaps yes by epsilon delta argument but I don't know if I am justified in doing ...
0
votes
2answers
31 views

maps from infinite sets to infinite sets

I know that the set of irrationals is uncountable, but I feel that there can always a bijection from one infinite set to a subset of another infinite set. Does this sound right? Say, from Irrationals ...
1
vote
1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
2
votes
1answer
44 views

Does something that is injective, surjective or bijective imply that it is a function?

As the title says. Sorry it seems like a silly question but it's something I've been wondering because it seems like sometimes the word "function" is omitted, but other times it is included
0
votes
1answer
51 views

Which function should I choose?

I am writing a paper and looking for a function.Its shape is just like this: when its x value is very small ,its y is close to 0 , but when x value is a little big, then its y value is very close to ...
0
votes
3answers
35 views

Function with limit constraints

In working with predator-prey population modeling, I ended up requiring a function $f(x)$ that satisfies the following conditions: $f$ is real and continuous, and so is $\frac{df}{dx}$; ...
1
vote
0answers
34 views

Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
0
votes
0answers
38 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...