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I'm reading Serge Lang's (S.L) linear algebra book. In the beginning, at function spaces section there is such a text: Let $S$ be a set and $K$ a field. By a function of $S$ into $K$ we shall ...

Here Why is Euler's Gamma function the "best" extension of the factorial function to the reals? there are several other functions beyond the common integral definition of the gamma ...

Prove that $$\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2^2}{2^3}}\bigg\rfloor+ \ldots$$ $x\geq{0}$ How should I ...

There is an exam question I tried to make up to test myself on trigonometry, but I am now confused... An example of the style of question is: The function f is defined such that $f(x) = 10 \cos x ...

Prove that, $$\bigg\lfloor{\frac{x}{n}}\bigg\rfloor=\bigg\lfloor{\frac{\lfloor{x}\rfloor}{n}}\bigg\rfloor$$ where $n \in{\mathbb{N}}$ My Attempt: Let $x=nt$. Then, I need to prove, $$\bigg\...

Suppose there are two smooth injective functions $$f:\mathbb R^n \rightarrow \mathbb R^p \text{ and } g:\mathbb R^m \rightarrow \mathbb R^p, $$ where $m\neq n$. The domains of $f$ and $g$ are ...

I was trying to define a function \begin{align*} f: \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3 \setminus \lbrace 0 \rbrace \end{align*} such that for any two vectors v and w, \...

Given the following sequence, $$1,1,2,3,5,8,3,1,4,5,9,4,3,7....$$ $$a_{n+2}=(a_n+a_{n+1}) \mod{10}\;\;\;\; \forall\;\;{n\geq{1}}$$ Prove that it is periodic? My Attempt: There can be atmost ...

I keep failing ungracefully at a seemingly simple task of determing a function to describe the coherence between two measured variables (a, b) and and a resulting value. Here's a complete round of ...

"What is a function?" can be answered as "Single-valued relations are called functions". But how can "What are the multi-valued function?" be answered? Will someone clarify my doubt why multi-valued ...

I am trying to find (point of touch) of two function 1.$g(x)=e^{-x^2/c}$ (gaussian) 2.$f(x)=x^2+4x+4$ (quadratic) I approached by equating tangents of both equation to equal.Thus I equated ...

In order to find the inverse of the function $y = x^3$ where $y = f(x) = x^3$ we need $x = f^{-1}(y)$, which we compute it as $x = y^{\frac{1}{3}}$ so the inverse function. But how do I calculate ...

I've been playing around with function $$f(x)=\sqrt{\frac{x+2}{x+1}}$$ I tried to find its domain, and I did so with finding the interval in which this applies $${({x\geq-2}\wedge{x\gt-1})}\vee{({x\...

prove that $x∉ℤ⇒\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$ So, I have that $(\lfloor{-x}\rfloor=n)⇔(n≤-x<n+1)$ And I also have that $(-\lfloor{x}\rfloor-1=n)⇔(\lfloor{x}\rfloor=-n-1)⇔(-n-1≤x<-n)$...

Let $f(x) = x^2 - 9x- 10$ We can state that $f(x) = (x + 1)(x-10)$ since I simply factored it. The roots of this function is $-1$ and $10$. However, what is the relationship between a factored ...

The requirements are: $f(x, y) = f(y, x)$ $f(x, x) = x$ $f(x, y) = f(x, x + y)$ f: $\mathbb{N}^2 \rightarrow \mathbb{N}$ I think $\gcd(x, y)$ works, but haven't found any other solutions nor have I ...

If you have a point list for $f(x)$, how do you determine the point list for $f(nx)$? The way I would have done it would be to plot the points for $f(x)$, identify the function and calculate the ...

How do I go about it. As I know that radical function doesn't apply to cube root

In order to apply the ideas of vector spaces to functions, the text I have (Wavelets for Computer Graphics: Theory and Applications by Stollnitz, DeRose and Salesin) conveniently says Since ...

Can you help me solve this question? More specifically, I want to try and solve this question using the reciprocal of the above fraction, but I don't know what that does to the question. Can you ...

Let $A$ be the set of all subintervals of $[0,1]$. For any function $f:[0,1]\times [0,1]\rightarrow A$, define $g(a,b)=[0,1]\setminus f(a,b)$. What are all functions $f$ such that for any $a,b\in[0,1]$...

I'm trying to generate a seemingly random list of integers without the use of a computational device; one capable of being unraveled with a mathematical function. What's the best way to devise a ...

Let f, g and h be the following functions. $f:Z \rightarrow \{-1,1\} \text{ defined as } f(x) =\begin{cases} 1, & \text{if $ x $ is even} \\ -1, & \text{if $ x $ is odd} \end{cases} $ $ g:...

As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on. I believe the answer is 0....

Question as follows Find the natural domain of $\frac f g$ $f(x) = \sqrt {-x} $ $g(x) = -(x+3)^2$ My attempt so far: $\frac {\sqrt {-x}}{-(x+3)^2}$ I am unsure of what you would do other than ...

I keep seeing $\max ( \cdot ) $ and $ \min ( \cdot ) $ everywhere. EDIT : It appears like this, $\max(4,9]$ or $\min[a,b)$ or sometimes with curly brackets, $\max \{0,100 \}$ etc. I ...

So , I missed one class of set theory and my instructor taught us something related to Hom- functor that I could not understand or find on the internet . He just wrote three lines : $$ Hom(A,B)=\...

If a circular arc of radius $1$ subtends an angle of $x$ radians . The centre of the circle is $o$ and the point $c$ is the intersection of two tangents lines at $a$ and $b$ . Now let $T(x)$ be the ...

Imagine a car travelling on a straight road at speed $u$ metres per second. The driver sees a kangaroo ahead and brakes to stop, with a reaction time of two seconds. In such circumstances, the ...

I am trying to solve the equation $f(x+y)-f(x-y)=2f'(x)f'(y)$ for all $f:\mathbb{R}\to\mathbb{R}$ non-constant, differentiable functions. Here is my progress: Any solution must be an even function ...

Can someone help me find a lower bound to the function $$\ln(1 - 0.5^u), u\ge0$$ I tried to use the Taylor formula, but it didn't seem to work. I would be grateful if anyone could give me some ...

I want to state a particular claim such as "functions in the form $f(x)=ag(x)^2-b$ have a solution in the form of $x=g^{-1}( \pm \frac{ \sqrt{f(x)+b}}{ \sqrt{a}})$ People know that not literally ...

I'm struggling with the following homework question: Let $n\in \mathbb{N}$. Prove that the function $b:\begin{Bmatrix}0,1 \end{Bmatrix}^n\rightarrow\begin{Bmatrix}0,1,2,...,2^n-1\end{Bmatrix}...

$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as the ...

Can you please help me find the proofs for the values of $k$ being real and distinct? Every time I tried to solve this equation, the answer I got had no real roots.

We know for$ f(x) = \arctan(x)$, $f'(x) = \dfrac1{x^2 + 1}$. Now we can write $x^2 + 1$ as $(x - i)\cdot(x + i)$ where $i = \sqrt{-1}$. So we can express $\dfrac1{x^2 + 1}$ as $\lambda\cdot\left(\...

Given that $x=1$ is the root of $f(x)=0$ where $$f(x)=x^6+a_1x^5+a_2x^4+x^3+a_2x^2+a_2x+1$$ and also given that $f(x+1) \ne 0$ Find Maximum number of distinct real roots $f(x)=0$ can have? My Try: ...

How do the recursive function for $\mod 5(x) = 0$ rest of division of $x$ by $5$. $$\begin{align} \mod&5(5) = 0\\ \mod&5(6) = 1\\ \mod&5(7) = 2\\ \mod&5(8) = 3\\ \mod&5(9) = 4\\ \...

So I have been thinking about if you could possibly create a function which always returns the positive of a number. To clarify, say I input $-4$, this function will output $4$. $$f(-5)=5$$ $$f(5)=...

Let $A$ be a Banach algebra and $I$ be a closed two sided ideal of it. Is the quotient map $\pi:A\to A/I$, a closed map?

Consider $P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9$, where $a$; $b$; $c$; $d$ are real. If the roots of $P(x)$ are in arithmetic progression, find the value of $a$. Although I am sure that this ...

If: $$f \left(\sqrt {4x}+6\over {3}\right)=\frac{x-9}{36}$$Then $$f(4\sqrt x)=?$$ I don't know how to use the first equality and then go for the second. Any hints?

If a function $f(x)$ is continuous and increasing at point $x=a,$ then there is a nbhd $(x-\delta,x+\delta),\delta>0$ where the function is also increasing. if $f' (x_0)$ is positive, then for $x$ ...

Under what minimal conditions must the range of a definable, injective function $\omega^{<\omega}\to\Bbb N$ have positive density in the integers? I'm very inexperienced in such matters but it ...

Here's my approach: express the sum as a telescoping series although I am not sure how to go about it I am sure it is either 29 or 31. Could someone help me?

There are two variables a and b over time t .For example ...

I have a function of two real variables which is given by the transformation rule $$f(x,y)=\frac{y}{1+x^2+y^2}.$$ I have to find the domain of $f$ which consists of all points $(x,y)$. When I examine ...

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(s+t)$ is also an integer. This is a problem ...

Does a function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}(x)=1-f(x)$ exist? Edit: I tried with different “simple” forms of function ($ax+b$, polynomes, $ae^{bx}+c$) without any inside as where to ...

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