All Questions
Tagged with real-numbers functions
183
questions
0
votes
2
answers
76
views
Prove that function is non-monotonic and is invertible
We have $$f(x)=\begin{cases}\frac x2,&x\in\Bbb Q\\-\frac x2,&x\in\Bbb R\setminus\Bbb Q\end{cases}$$
Prove that function is non monotonic and is invertible.
I tried to take points from $\Bbb ...
user873532
0
votes
3
answers
72
views
{$\frac{1}{x}$} , $0<x\leq1$ in terms of {$x$}, $x\geq 1$, {$x$} is fractional part of $x$
$f(x)$ $=$ {$\frac{1}{x}$}, $0<x\leq1$ where {$x$} denotes the fractional part of $x$.
$g(x) =$ {$x$}, $x\geq 1$
I want an expression for $f(x)$ in terms of x and $g(x)$.
My try-
If $x\in \mathbb{Z}...
user826419
0
votes
1
answer
47
views
Prove that $\left | f(x) - a \right | \leq \frac{1}{2} \left | x - a \right |$
$f(x) = 1 - \frac{1}{x}(\sqrt{1 + x^2}-1)$
$|f'(x)| \leq \frac{1}{2}$
$a$ is a solution for $f(x) = x$ where $0.65 < a < 0,7$
The question says:
Prove that $\left | f(x) - a \right | \leq \frac{...
- 433
7
votes
2
answers
312
views
What is a condition for two real functions $f,g$ to "commute", so $f(g(x))=g(f(x))$?
Say I'm given two functions $f,g$. Can I tell if they "commute" without actually trying them in the formula
$f(g(x))=g(f(x))$?
And given a function $f$, is there a way to find all functions $...
- 405
3
votes
1
answer
67
views
How do I solve this in an understandable and direct way? [closed]
For each $i \in \Bbb N$, let $f_i: \Bbb N \mapsto \{0, 1\}$.
Let $A = \{f_i : i \in \Bbb N\}$ and $E = \{n \in \Bbb N : f_n(n) = 0\}$.
Does there exist a $f \in A$ such that $E = \{n \in\Bbb N : f(n) =...
user850289
2
votes
2
answers
377
views
Is a circle a multivalued function?
I don't really understand multi-valued function. I hope one of you can make me understand it. What I've learned from google, I suppose that a multi-valued function is a binary relation that maps the ...
- 2,200
3
votes
1
answer
81
views
Suppose $\sum_{n\ge 1} |a_n| = A<\infty.$ Under what conditions is $\sum_{n\ge 1} \epsilon_n a_n = [-A,A]$, for $\epsilon_n \in \{-1,1\}$?
Consider the space of sequences:
$$
\mathcal{E} = \{\{\epsilon_n\}_{n= 1}^{\infty}: \epsilon_n = \pm 1\}
$$
This can be considered a "random choice of sign" in the probabilistic context, for ...
- 8,389
0
votes
3
answers
94
views
Is it true that there is a bijection $[0, 1) \to \mathbb{R}$?
Is there is a bijection from $[0,1)$ to $\mathbb{R}$?
I thought of an instance, $$\frac{\sqrt{x(1-x)}}{x-1}.$$
- 127
1
vote
2
answers
436
views
Find all real solutions $x$ for the equation $x^{1/2} − (2−2x)^{1/2} = 1$
This is what the answer says:
Note that the equation can be rewritten as $\sqrt{x} − \sqrt{2 − 2x} = 1$,
and the existence of such real $x$ implies that $x$ is larger than or equal to $0$ and $x$ is ...
- 67
0
votes
1
answer
179
views
Paring function - Output becomes exponential for big real inputs
I am using a Cantor pairing function that takes two real number output unique real number.
def cantor_paring(a,b):
return (1/2)*(a+b)*(a+b+1) + b
This work ...
- 101
3
votes
2
answers
381
views
What does $f:\mathbb R \rightarrow \mathbb R$ mean?
This is simply a basic notation question: what is the meaning of
$$f:\mathbb R \rightarrow \mathbb R$$
I imagine it's some sort of function to do with the set of real numbers, perhaps some sort of ...
- 6,550
10
votes
3
answers
1k
views
Significance of Codomain of a Function
We know that Range of a function is a set off all values a function will output.
While Codomain is defined as "a set that includes all the possible values of a given function."
By knowing ...
- 386
-2
votes
2
answers
116
views
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$? [closed]
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$?
If so, any thoughts on how to prove this?
- 431
0
votes
1
answer
135
views
true or false- continuous functions
I'm having some hard time deciding if those sentences are true or false:
$1$. If $f$ is continuous on $\mathbb{R}$ then if $\left|f(x)-x\right|<1$ for every $x$ on $\mathbb{R}$ then $f$ is getting ...
- 65
0
votes
1
answer
41
views
Explain a confusing bound for the integral of a decreasing function.
I am reading a solution of an exercise. In the solution, it says the following:
Consider $g(x,t):=\frac{x}{(1+tx^{2})t^{\alpha}}$, where $x\in (0,\infty)$, $t=1,2,3,\cdots$ and $\alpha>\frac{1}{2}$...
- 5,930