All questions with tag [math: functions]


why function over just a relation?

Is there a the real world instance revealing information that creates a relationship is better than one that creates simply a relationship? what the real world circumstance encourages the added problem on relationship so we have functions? Thanks.
2022-07-25 17:47:21

Difference in limits because of greatest-integer function

A Problem: \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} causes the remedy: $1$ But the very same function confined in a best integer function causes a $0$ \begin{equation}\lim_{x\to 0} \left\lfloor{\frac{\sin x }{x}}\right\rfloor\end{equation} Why? My ideas: ยข [The value of the first function often tends to 1 as a result of t...
2022-07-25 17:47:02

One sided limit of an increasing function defined on an open interval

Let $f:(a,b)\to \mathbb{R}$ be a purely raising function. Does the restriction $\lim_{x\to a^+}f(x)$ always exist and also is an actual number or $-\infty$? If so, is it real that $\ell=\lim_{x\to a}f(x)\le f(x) \ \ \forall x\in (a,b)$? Please give evidence.
2022-07-25 17:43:22

Proving $\cos \theta = \sin\left(\frac{\pi}{2} - \theta\right)$ for all $\theta$

By drawing a right triangle it is obvious that $\cos \theta = \sin\left(\frac{\pi}{2} - \theta\right)$. I'm trying to prove to myself that this is true for all values of $\theta$ by following the reasoning on where the sine graph is inverted vertically by plugging in $-\theta$ and then shifted to the right by $\frac{\pi}{2}$ to get the cosine g...
2022-07-25 17:42:42

Finding range of $f(g(h(x)))$

$$\begin{align*} f(x) &= \frac{2}{x+1}, \\ g(x) &= \cos x, \\ h(x) &= \sqrt{x+3} \end{align*} $$ Find the series of $f(g(h(x)))$. Please clarify the trouble.
2022-07-25 17:41:29

Who came up with the arrow notation $x \rightarrow y$?

I read that the arrowhead notation $x \rightarrow y$ was designed in the 20th century. That presented it? Each map requires both a specific domain name and also a specific codomain (not simply a domain name, as in previous solutions of set concept, and also not simply a codomain, as in type concept). - - Lawvere and also Rosebrugh Sets for Mat...
2022-07-25 17:22:51

Verifying a proposition on image and preimage: $f(A\cap B)\subseteq f(A)\cap f(B)$ and $f^{-1}(C\cap D)=f^{-1}(C)\cap f^{-1}(D)$

I need help in verifying the following please: Let $f:X\to Y$ be a function, and let $A,B\subseteq X$, and let $C,D\subseteq Y$. Then $f(A\cap B)\subseteq f(A)\cap f(B)$ $f^{-1}(C\cap D)=f^{-1}(C)\cap f^{-1}(D)$ I am not quite certain how to get started, and would appreciate any help. Thanks. Edit: Please excuse my ignorance, but I think...
2022-07-25 17:20:44

Proof that infinite functions can fit a table of numerical values

Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal digits of accuracy). Like the table shown here: where constant1, constant2, etc. are boundary conditions. My question is whether we can always find...
2022-07-25 17:17:39

Can the output of a mathematical function be another mathematical function?

Apologies if this is actually mathematical gibberish. I'm very familiar with mathematical functions at a simple level. A function relates a set of inputs to a set of outputs. What I'm trying to denote is a function, G, which, when given a set F and a true/false value, $\delta$, returns another function which has $n$ as a free variable. For refer...
2022-07-25 17:06:26

Find a function that fits data and has certain characteristics

I have some information. data = [ (10000, 0), (100000, 0.25), (10000000, 0.5), ] I intend to locate function (s) suitable this information. I have a feasible beginning point: f(x) = Ax / (B + x) - C Can I locate A, B, C pleasing the information? The function needs to be continual. f(x) for x<=10000 does not matter yet need ...
2022-07-25 16:58:09

Mathmatical representation of recursion function

Well i'm not so efficient mathematics, yet i have the adhering to job: Below is the code: int foo(n): if n <= 0: return 1 else: return foo(n-1)+foo(n-3)-1 What is foo (7) will return? So i need to address without making use of any kind of tools. And also ideas concerning attracting trees by hand problems me. Exis...
2022-07-25 16:44:23

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$ (see for reference I am asking yourself if one can get wonderful depiction of $L_2$ - standard of the function $f^{\alpha}(x)$, particularly $$ \int_{-\...
2022-07-25 16:41:28

If $ f(3x-1)= 12x+5$, what is $\ x \circ f(x)$?

Here, I am sharing just an example problem which is given in one of my textbooks: $$ \ \large{ f(3x-1)=12x+5 \ , \\ x \circ f(x)= \, ? \ } \ $$ And, on below of the question, the book has shown an example solution for that: $$ \text{Instead of } \ \mathit{x}, \ \text{if we write the inverse function of } \ \mathbf{3x-1} \, \text{;} \\ \large{ ...
2022-07-25 16:33:50

function f satisfies f(xy) = f(x)/y , f(30) = 20. Find f(40)

The function $f$ pleases $$f(xy) = \frac{f(x)}y$$ and also $f(30) = 20$. Locate $f(40)$.
2022-07-25 13:30:39

How to combine ratings given by two different functions into one rating

I have a set of things: s1, s2, s3 . There are 2 functions that rank these things: f1() and also f2(). As an example f1 (s1) = 4. f1() returs values in [0,15 ], yet f2() returns values in [0,100 ]. In addition, the raiting offered by f1() is more vital than the one offered by f2(). Just how do I incorporate the raitings returned by f1() and a...
2022-07-25 13:19:38

Iterated Root Mean Square-Arithmetic Mean

Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible? if not possible, what is the closed form of it as known functions ? $$AGM=M(x,y)=\frac{\pi}{4}\frac{x+y}{K(\frac{x-y}{x+y})}$$ where $K(m)$ is the complete elliptic integral of the first kind: $$K(m)=...
2022-07-25 13:14:47

Inverse of $f^{-1}(x)=x^5+2x^3+3x+1$ question

Let $f$ be a one - to - one function whose inverted function is $f^{-1}(x)=x^5+2x^3+3x+1$. Calculate the value of $x_0$ such that $f(x_0)=1$. I am perplexed regarding what this inquiry is asking me, specifically given that I do not recognize the under the $x$ variable.
2022-07-25 13:09:54

calculating the amplitude of a cosine function

I intend to have the ability to have the ability to get the amplitude of the adhering to function: $$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$ I am searching for a means to get the amplitude of this function. This is generally straightforward for when there is one value of $\omega$ to take into consideration...
2022-07-25 12:53:35

Finding the Inverse of a Summation

I have actually seen extra details variations of this inquiry yet my inquiry is extra basic. For any kind of offered summation does there exist an inverted. Otherwise, just how does one inform if the function has an inverted. Do these inverses constantly have shut kinds (I visualize they do not)? Just how can one inform when a function such as a...
2022-07-25 12:50:41

How to check if function is Lipschitz continuously differentiable?

I do have a problem with achieving convergence in Newton method (using Armijo rule) for system of algebraic non-linear equations. I suspect that my function is not continuously differentiable, however I'd like to be sure if that is so. How do I test it if my F( x ) is Lipschitz continuously differentiable? Thanks in advance, Regards
2022-07-25 12:47:30