0 functions questions.

Proving that function space is a vector space over field

0 answers, 30 views linear-algebra functions vector-spaces
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 ...

-1 I still don't understand why we don't use other definitions of the gamma function.

1 answers, 54 views functions definition factorial
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 $\lfloor{x}\rfloor=\bigg\lfloor{\frac{x+1}{2}}\bigg\rfloor+\bigg\lfloor{\frac{x+2}{2^2}}\bigg\rfloor+\ldots$ [duplicate]

0 answers, 32 views functions floor-function
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 ...

Natural domain for $f(2x)…$

1 answers, 23 views functions trigonometry

2 Find the limit of $x(x + 1 - \sin(\frac{1}{1+x})^{-1})$ as $x \rightarrow \infty$

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....

1 Natural domain of $\frac f g$

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 ...

1 What is $\max$ and $\min$ in set theory?

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 ...

4 Find all polynomials $P(x) \in \mathbb{Z}[x]$ such that if $P(s)$ and $P(t)$ are both integers, then $P(s+t)$ is also an integer

1 answers, 58 views functions polynomials contest-math
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 ...

6 A function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}(x)=1-f(x)$?

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 ...