All Questions
Tagged with real-numbers functions
183
questions
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60
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Functions problem from CGMO 2010
Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $...
- 81
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1
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823
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Does $f:\mathbb{R} \rightarrow \mathbb{R}$ mean that $f(x)$ is defined for all real inputs?
$f:\mathbb{R} \rightarrow \mathbb{R}$ basically means that the domain of $f$ is the set of real numbers and the range of $f$ is the set of real numbers. However, does it mean that $f(x)$ is defined ...
- 3,186
1
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0
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153
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What does trigonometric functions of real numbers really mean?
According to right triangle definition, trigonometric functions relates the angle to the ratio of sides of triangle. These functions take angles as input.
According to unit circle definition, ...
6
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2
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367
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How to strengthen $ h \big( 2 h ( x ) \big) = h ( x ) + x $ to force $ h $ to be linear?
Let $ h : \mathbb R \to \mathbb R $ be an injective function such that
$$
h \big( 2 h ( x ) \big) = h ( x ) + x
$$
for all $ x \in \mathbb R $, and $ h ( 0 ) = 0 $. What would be an as mild as ...
- 12.1k
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1
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41
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What is the correct inverse function for $f(x) = x^2$. Question about terminology.
Is the inverse function $g_1(x)=\sqrt{x}$ or is it $g_2(x)=-\sqrt{x}$. With what terminology can we describe both of these functions?
- 2,527
1
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0
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17
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Looking for a sigmoid-like function with different properties
I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
- 3,439
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1
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31
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How to Quantify Utility/Pleasure/Pain using the Positive Real Numbers?
I am studying about Cardinal Utility in Economics (or more generally, how to quantify pleasure and pain!)
Intuitively, I assign a positive number to pleasurable experiences, and a negative number to ...
1
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2
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72
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Prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
$$g(x) = \ln(x + 1 + e^{-x})$$
My question is prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
How do I do that?
My attempts:
I have only successfully proved the ...
- 433
1
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1
answer
114
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Can construct a bijection between R - Q and ( (R - Q) ∩ [0,1] )?
I've tried to show that:
$$[0,1]\sim([0,1] ∩R-Q)$$
I know from this answer :
$$[0,1]\sim R-Q$$
But how to construct a bijection between R-Q and $([0,1]∩R-Q)$ ?
I think the function would be like $f:R-...
- 382
0
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1
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167
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Will the roots of $p(p(x))=0$ be purely real or purely imaginary or neither?
Question
The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x))=0$ will have -
a) only purely imaginary roots
b) all real roots
c) two real and ...
- 175
1
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1
answer
99
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If $f$ is a function such that $f(f(x))=x^{2}-1$ determine the function $f(f(f(f(x))))$
I have tried $f(f(f(f(x))))=f\left(f\left(x^{2}-1\right)\right)$ . Since we know that $f(f(x))=x^{2}-1$, we have
$$
\begin{aligned}
f\left(f\left(x^{2}-1\right)\right) &=\left(x^{2}-1\right)^{2}-1 ...
- 55
2
votes
1
answer
171
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Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
1
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0
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70
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Is this a surjection from $(0,1) \rightarrow \mathbb{R}$?
I am trying to think of creative bijections from $(0,1) \rightarrow \mathbb{R}$ that can serve as a proof of the fact that the cardinality of the interval $(0,1)$ is equal to the cardinality of $\...
- 1,186
3
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3
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211
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From which set does the number $\sqrt[3]{-1}$ belong to?
I was trying to draw the function $f(x)=\sqrt[3]{x^2(6-x)}$ by hand (I'm in my first year of engineering; having Calculus I; this drawing is actually an exercise given for my class) and used ...
2
votes
2
answers
53
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what are all functions with $x>1$ and $y>1$ $\rho$ that follows $\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$ and is continuous
What are all functions with $x>1$ and $y>1$ $\rho$ that follows $$\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$$ and is continuous
If this doesn't have any solutions then prove no such ...
user872441
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2
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76
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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 $\Bbb ...
user873532
0
votes
3
answers
72
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{$\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
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1
answer
47
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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
314
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
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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
381
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 ...
- 2,209
3
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1
answer
81
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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 ...
- 8,369
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
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2
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441
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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
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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
385
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 ...
- 6,560
10
votes
3
answers
1k
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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
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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
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135
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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
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1
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42
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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