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

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0
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1answer
19 views

Solving functional equation $f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n).$

Given natural number $n, k$ consider nondecreasing function $f:\mathbb{N}\cup {0} \to \mathbb{N}\cup {0}$ such that $$ f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n), $$ for ...
2
votes
1answer
30 views

Could you give an example of an injective function $f:\mathbb{Z_+}^n\rightarrow \mathbb{Z_+}$ for an integer $n$ s.t. $2\leq n$?

We know that both of the domain the the co-domain are countable sets, so there is a bijection between them, Is there any SIMPLE injection? Here is some injection which I thougt of, but It turns out ...
1
vote
1answer
74 views

$f(AB)=f(A)f(B)$, show that $f$ is or injective or zero

Let $f\in\mathcal{L}(\mathcal{M}_n(\mathbb{R}))$ such that: $\forall(A,B)\in\mathcal{M}_n(\mathbb{R}),f(AB)=f(A)f(B)$ How can I show that $f$ is or injective or the null function ? What I have ...
0
votes
0answers
25 views

Using function composition, show that $f(x)$ and $f^{-1}(x)$ are inverses of one another.

Consider the function $$f(x)=\frac{1}{3}x+2.$$ a) Find the inverse of $f(x)$ and name it $g(x)$. Show and explain your work. b) Use function composition to show that $f(x)$ and $g(x)$ ...
1
vote
1answer
27 views

Derivative Functions [on hold]

Consider $f(x)= ax^2 + bx$ where $a$ and $b$ are real numbers. If $f(1)=-1$ and $f'(-1)=-7$, find the values of $a$ and $b$? I genuinely do not understand how to do it! Please help
1
vote
1answer
42 views

How many functions are there from 5 to 0?

I have just learned that from a set of $n$ element to a set of m elements, the number of functions is $m^n$. However, how about from $5$ to $0$? $5$ is a natural number which can also be considered as ...
-1
votes
0answers
24 views

Can someone help me with this math problem [on hold]

Can someone help me with this math problem? for $f(t) = 0.16t^2 - 1.6 t + 35$ find and simplefy $\Delta f / \Delta t$ . for the intervals $[b,5]$ i guess delta means change in this context, not ...
0
votes
1answer
16 views

Discontinuities with an oscillating function in the denominator

The problem is to find the discontinuities in the following function \begin{equation} f(x) = \frac{4x+1}{5cos(\frac{x}{2})+1} \end{equation} I know the function will be discontinuous whenever ...
1
vote
2answers
48 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
2
votes
1answer
21 views

How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
0
votes
0answers
47 views

Prove that the minimum of this function is less than $1$

Here is the function: $$D^K(\theta)= 2\frac{\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-\theta)^{2K-i}}{\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-2\theta)^{K-i}}$$ Show that for ...
0
votes
0answers
17 views

Help with understand the growth order of functions

I am taking an Algorithms class and I understand everything that relates to the asymptotic growth and Order of growth for a given function (Theta, Omega, etc). However, I am having trouble in ...
0
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0answers
10 views

Empirical Intensity Function

I would like to ask for help determining what other ways are there to compute the "empirical intensity function" of a process. In essence, given that I observe the occurrences of an event in time ...
0
votes
0answers
10 views

Is it possible for the derivative of a multivariate function to be a function of lesser dimension?

Let's say I have some function $f$ such that $f'(a,b,c,d)$ exists for all $a$, $b$, $c$, and $d$, and that $f(a,b,c,d)$ is dependent upon $a$, $b$, $c$, and $d$. (That is, $f(a,b,c,d)$ can't be ...
0
votes
1answer
18 views

How do find out if a piecewise function has a maximum or minimum/how many?

Given a piecewise function with the domain $[0,2]$ and $f(x) = \begin{cases}2x^3 - x^2& \text{if } x \leq1\\ \frac{x+1}{x-1}& \text{if } x<1\end{cases}$ How does one decide anything ...
5
votes
1answer
27 views

newbee question: How to create a function?

I wonder how to create a function based on the characteristics. suppose I have function $f$ and $g$ like this: $f(x,g(x,y,z)) = y$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,$ $g(x,f(x,z),a) = z$ With ...
0
votes
0answers
32 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
0
votes
1answer
25 views

Piecewise function and maximum/min

$f$ is given by $$f(x) = \begin{cases} x^2 \cdot \sin(x) & \text{if }x\geq0\\ 1/x&\text{if } x < 0 \end{cases} $$ and if we look at the interval $[1,2]$ Given the above, how can we use ...
0
votes
2answers
27 views

What is guaranteed given these conditions?

If $f'(0) = 0$, $f(-1) = -1$, $f(1) = 1$, then what is guaranteed to be true? I know the answer, it's ($2$), but I am interested in knowing why this is the case, and why the other's aren't true ...
12
votes
4answers
146 views

How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $?

Suppose that $f(x)$ is a bounded continuous function on $\mathbb{R}$,and that there exists a positive number $a$ such that $$f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $$ is constant. ...
1
vote
1answer
11 views

composition sum of functions/sum of composition of functions

I know it sounds really dumb, but is it true that $(f_1+f_2)\circ g=f_1\circ g+f_2\circ g$? I know it must be really elementary, but I don't recall seeing this being proved (or defined) explicitly.
0
votes
1answer
29 views

excel- function with multiple variables

So I'm not good with excel (computers in general) and can do some things but this one is out of my league. This is the problem: The cost of a used car is highly correlated with the following ...
1
vote
1answer
20 views

Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
1
vote
0answers
26 views

Ordinal arithmetic and functions

I have two function $G$ and $F$ defined on ordinals and I know that $$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
0
votes
0answers
37 views

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,…$. Prove that $f=0$ on $[0,1]$.

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,...$ $$ \int_0^1 f(t)t^n dt=0$$ Then prove that $f=0$ on $[0,1]$. Here I don't want the proof. I have one proof. But I have ...
0
votes
1answer
32 views

Is there a mathematical function that converts two numbers into one so that the two numbers can always be extracted again [on hold]

can you please explain the answer to question "Is there a mathematical function that converts two numbers into one so that the two numbers can always be extracted again" given at ...
0
votes
1answer
26 views

Why is this function from a polynomial to $\mathbb{Z}^{i+1}$ injective?

Part of my professor's solution: Let $A$ = the set of all polynomials with integer coefficients. Let $A_i$ be the set of all polynomials of degree $i$ with integer coefficients. Then $A = ...
0
votes
4answers
39 views

How to show if a function f is open/not open

could you help me out? Let the real function $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=x^2$. Show that $f$ is not open. How do I go about doing this? Thanks in advance!
0
votes
0answers
23 views

How to prove that a function is compact (closed and bounded)?

The specific function I am looking at is $f(x_1,x_2) = x_1x_2 + \frac 1{x_1} + \frac 1{x_2}$, where for a fixed $a > 0, f(x) \le a$ and $(x1,x2) > 0 $ I'm really just looking for where to ...
1
vote
2answers
129 views

Can there be variations on the Witch of Agnesi?

The function known as "The Witch of Agnesi" can be constructed using a circle of radius $a$, and is written in Cartesian coordinates as $$f(x)=\frac{8a^3}{x^2+4a^2}$$ The family of functions that ...
0
votes
0answers
5 views

Condition on Vector Boolean Function to be Bijective

Suppose the vector boolean function be $$\begin{align} f:F^n_2 \longrightarrow F_2^n \\ (x_1,\dots ,x_n) \longrightarrow (x_2,\dots x_n,g) \\ \\ g:F^n_2 \longrightarrow F_2 \\ (x_1,\dots ,x_n) ...
0
votes
2answers
31 views

Why is $f$ doesn't have the intermediate value property?

$f$, defined on $(0,2)$ doesn't have the intermediate value property: $$f(x) = \begin{cases} x &\mbox{if } x \in (0,1) \\ 0 & \mbox{if } x \in [1,2) \end{cases} $$ Whereas, $g$ defined on ...
0
votes
0answers
16 views

Finding a function with desirable behaviour

I was playing around with math in my spare time and came up with the following problem. Suppose I have a $1$-parameter family of lines in the plane. Each line is given by the following equation ...
1
vote
1answer
101 views

Is sin(1/x) periodic?with what period time? [on hold]

Is sin(1/x) periodic? with what period time?
0
votes
2answers
61 views

Show that $\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}},\cdots$ [on hold]

Show that $$\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}},\cdots$$ is increasing. Thanks for any help.
2
votes
0answers
44 views

Sign of a Function over $[0,\pi]$

Let $\lambda \in \mathbb{R} $ such that $\left| \lambda \right| \ne 1$ $$f_\lambda (x)=(λ\cos x−1)(λ−\cos x)$$ Show that : For $|λ|<1$, $$f_λ(x)=\begin{cases} <0 ...
0
votes
2answers
11 views

functions and statistics [on hold]

Find the values of the constants $k$ and $m$, if possible, that will make the function $f$ continuous everywhere. $$f(x)=\begin{cases} x^2+5 &, x>2\\m(x+2)+k &, -2<x\le2 \\2x^3+x+7 ...
0
votes
1answer
47 views

Use induction to prove that a function is not one to one

Suppose that m and n are positive integers with m > n and f is a function from $\{1, 2,\ldots, m\}$ to $\{1, 2, \ldots , n\}$. Use mathematical induction on the variable n to show that f is not ...
1
vote
1answer
32 views

Use the implicit function theorem to determine when can the following equation be solved:

I was asked to determine when the equation $f(x,y)=y^2+y+3x+1=0$ for $y$ in terms of $x$. First the I was asked to provide and answer without using the Implicit Function Theorem (IFT), so I simply ...
1
vote
3answers
74 views

How is $\sin x$ considered a function

Yeah, it's a simple question, I forgot the exact reasoning behind it. also would a one-to-one function be considered a function (obviously yes)? If it is then how come $\sin x$ is a function. I'm ...
1
vote
2answers
28 views

Stuck with finding the domain of a function with a logarithm

Find the domain of the function $$g(x)=\log_3(x^2-1)$$ This is what I got so far: $$\{ x\mid x^2-1>0\} =$$ $$\{ x\mid x^2>1\} =$$ $$\{ x\mid x>\sqrt { 1 } \}= $$ I don't know where to ...
0
votes
1answer
18 views

Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me ...
0
votes
0answers
9 views

A cell dies at a constant rate r and what is the density function of its life time.

The problem: A cell dies at a constant rate $r$ and the its life time is the duration from t=0 to when when it dies. what is the density function of its life time $l$? I have done some relevant ...
0
votes
1answer
41 views

Use complete induction to prove the following

Let $f:\Bbb N\to\Bbb N$ be given by $$f(n) = \begin{cases} 3, & n = 1 \\ 1, & n = 2 \\ 2f(n-1)+f(n-2), & n \ge 3 \end{cases}$$ prove that for $f(n)$ for all is odd all natural numbers ...
1
vote
1answer
17 views

Transient and Steady-State Response

What is the transient and steady-state response in the next equation: $$2 + 5t + 3\exp(-0.1t)$$ I am looking to understand how to identify both responses just looking the equations....Please Help ...
0
votes
3answers
51 views

Is it true that every injective function must be surjective? [duplicate]

I believe it is false, because an injective function never maps elements of the domain to the same element of its codomain, where as the surjective function can map an element of the codomain to any ...
0
votes
1answer
19 views

How to find number of disctinct functions from set A to set B

Let's say there is set A {1, 2, 3} and set B {a, b} While, I know that to find the total number of functions, it's just number of elements from B ^ number of elements from A But I just don't ...
-2
votes
0answers
13 views

I need help with boolean functions [on hold]

Can someone write Boolean functions for these four schemes?Schemes, please ??? Thank you in advance.
1
vote
1answer
19 views

Determine the $n$th string among those of a given length in alphabetical order starting at a given string and using a given character set?

When building strings using a particular character set (the set can change), such as in a brute-force password cracking, how would I determine which string occurs in the $n$th position when the ...
2
votes
1answer
36 views

A question on the Wronskian

Let $f(z),g(z)$ be two complex-valued functions defined in some domain $D$. Suppose we want to show that $$f(z)+g(z)\neq 0 \tag1$$ for all $z\in D$. I think I'm right in saying we can use the ...