All Questions
Tagged with real-numbers functions
183
questions
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43
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How to combine the $4$-dimensions of spacetime into 1 dimension?
I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...
3
votes
1
answer
73
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Whether the given function is one-one or onto or bijective?
Let $f:\mathbb{R}\to \mathbb{R}$ be such that
$$f(x)=x^3+x^2+x+\{x\}$$ where $\{x\}$ denotes the fractional part of $x$. Whether $f$ is one-one or onto or both?
For one-one, we need to show that if $f(...
- 4,500
1
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1
answer
127
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$g(f(x + y)) = f(x) + (x+y)g(y).$ Value of $π(0) + π(1)+\dots+ π(2024)$?
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$:
$$g(f(x + y)) = f(x) + (x+y)g(y).$$
The value of $π(0) + π(1)+\ldots π(2024)$ is?
I found the question on Mathematics Stack ...
- 55
6
votes
1
answer
168
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Prove that for any function $f:\mathbb R\to\mathbb R$ there exist real numbers $x,y$, with $x\neq y$, such that $|f(x)-f(y)|\leq 1$.
I need help with a 9th grade functions exercise:
Prove that for any function $f:\mathbb R\to\mathbb R$ there exist real numbers $x,y$, with $x\neq y$, such that $|f(x)-f(y)|\leq 1$.
I tried assuming ...
- 81
1
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1
answer
34
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A claim regarding some summations of monotonic functions in fraction
I am trying to prove this claim but it seems the math somehow does not work out...
Let $f_1,f_2,g_1$ and $g_2$ be real-valued, strictly positive, continuously differentiable and strictly decreasing ...
- 111
1
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2
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119
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Show that a function is $f$ bijective if $f(f(f(2x+3)))=x$ for all real $x$
Let $ f : \mathbb{R} \to \mathbb{R} $ is a function such that
$$ \forall x\in\mathbb{R} : f(f(f(2x+3)))=x $$
Show that $f$ is bijective.
We have to show that $f$ is injective and surjective.
How do we ...
- 113
4
votes
6
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682
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Is $x^3$ really an increasing function for all intervals?
I had an argument with my maths teacher today...
He says, along with another classmate of mine that $x^3$ is increasing for all intervals. I argue that it isn't.
If we look at conditions for ...
- 45
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1
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46
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How can I study the changes of $f_k(x)=\frac{e^{kx}-1}{2e^x}$
Let $f_k(x)$ be a function defined on $\mathbb{R}$ by
$$f_k(x)=\frac{e^{kx}-1}{2e^x}$$
Where $k$ is a real , How can I study according to the values of $k$ the changes of the changes of $f_k$
I ...
- 59
-3
votes
1
answer
100
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Is this an injection from $\mathbb{R}_+$ to $(0,1)$? [closed]
I am wondering if $f: \mathbb{R}_+ \rightarrow (0,1)$ is an injection if $f$ just moves the decimal point to the left of each number an equal amount of times as how far the decimal point is from the ...
- 2,979
0
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1
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26
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Is it possible for a function to be continuously derivable over its entire open domain except for a removable discontinuity?
For example, is there a function $f \in \mathcal{C}(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R} \setminus \{ 0 \})$ such that
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
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4
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67
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Rudin PMA 4.31 - does the elements of $E$ have to be ordered (smallest to the biggest)?
Here is a reformulation of Rudin PMA $4.31$ remark:
Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ ...
- 1,231
2
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2
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59
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Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
7
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2
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157
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Find all real functions that satisfy: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ $\in$ $\mathbb{R}$
I am trying to find all real functions that satisfy the property: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ real numbers. I tried to substitute $x$ and $y$ with $0$ but end up with nothing, then I ...
- 301
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1
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98
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Functional square root of a function $F([a,b])=[1βbr+a,1βbr+a+b]$.
Suppose we have a function $F\colon \mathbb{R}^2 \to \mathbb{R}^2$ over a numeric pair (pair of real numbers) $[a, b]$ such that:
$$F([a,b]) = [1-br+a, 1-br+a+b]$$
for some $r\in\mathbb{R}$.
find a ...
- 67
2
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2
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122
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Formula for the $n$-th term of the sequence $1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$, where $f_n := \frac{1}{n} (f_{2n} - f_n)$
I'm struggling with this sequence.
$$1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$$
Where, $f_n := \frac{1}{n} (f_{2n} - f_n)$
You can also work it out for negative powers of 2,
$$f_\frac{1}{2} ...
- 71
7
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2
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548
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Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ where $f(xf(y)+f(x)+y)=xy+f(x)+f(y)$
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ for two real numbers $x$ and $y$ where $f(xf(y)+f(x)+y)=xy+f(x)+f(y)$
For $x=0$ and $y=-f(0)$ then $f(-f(0))=0$. So, there is a real root $r_0$ ...
- 351
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0
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30
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Complex Functions Examples
I was asked to give an example of a function:
i) whose domain isn't equal to its codomain
ii) whose domain isn't equal to its image
iii) whose codomain isn't equal to its image
iv) a function $f$ from ...
user1071088
0
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1
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30
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Finding domain and range without equation
We are given that $f(x)$ has domain $x \geq -4$ and $f(x) < -1$. All numbers in $\mathbb{R}$.
Now we want to find the domain of $3f(x+1)+4$. My solution is $x+1 \geq -4$ so new domain is $x \geq -5$...
- 753
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2
answers
70
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Proof Surjective function with no ''given'' function
$f_1 : \mathbb{R}^2 \longrightarrow \mathbb{R}$ and $f_2 : \mathbb{R}^2 \longrightarrow \mathbb{R}$ and define $f: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ as $f(x,y) = (f_1(x,y),f_2(x,y))$.
...
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-1
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1
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44
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Does there exist a real-valued function such that for any proper subset of real numbers the function maps outsidr of the set?
Does there exist a bijective function $f$ from the real numbers to the real numbers such that for any non-empty proper subset of real numbers $R$ there exist $x$ in $R$ such that $f(x)$ is not an ...
- 1,104
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1
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76
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Proving that the map $f:\mathbb R \to \text{Seq}(\mathbb Q)/\sim$ is surjective
I was reading about constructing Real numbers using Cauchy sequences of rational numbers.
To be more specific, let $\text{Seq}(\mathbb Q)$ be the set of all Cauchy sequences of rational numbers and ...
- 4,962
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1
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108
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Show that $f(rx)=(f(x))^r$ for all $r\in \mathbb{Q}$ if $f(x+y)=f(x).f(y)$ for all $x,y\in \mathbb{R}$
$f:\mathbb{R}\to \mathbb{R}$ be function satisfying $f(x+y)=f(x).f(y)$ for all $x,y\in \mathbb{R}$ and $lim_{x\to 0} f(x)=1$ then show that $f(rx)=(f(x))^r$ for all $r\in \mathbb{Q}$.
Here is what I ...
- 322
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124
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Converse of Bolzano Weierstrass Theorem
Bolzano Weierstrass Theorem (for sequences) states that Every bounded sequence has a limit point.
However the converse is not true i.e. there do exist unbounded sequence(s) having only one real limit ...
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47
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Find $f(3)$ if $f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x$ [duplicate]
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
$$f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x$$for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$...
- 800
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1
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Show that $\mathbb{R}$ and the open interval $(-1, 1)$ have the same cardinality.
I am a little confused about using functions to show that two sets of intervals have the same cardinality.
I believe that if we can find a bijective function $f$ such that $f: \mathbb{R} \to (-1, 1)$, ...
- 115
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0
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37
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Bijection from $\mathcal{P} (\mathbb{R})$ to the set of functions from $\mathbb{R}$ to $\mathbb{R}$ [duplicate]
Iβm a bit confused as to how we get the bijection between a powerset of a set to the set of functions from that self to itself
I can see the obvious bijection from the powerset to the set ${[0,1]}^{R}$...
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4
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1k
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Why is the range a larger set than the domain?
When we have a function $f: \mathbb{R} \to \mathbb{R}$, I can intuitively picture that and think that for every $x \in \mathbb{R}$, we can find a $y \in \mathbb{R}$ such that our function $f$ maps $x$ ...
- 319
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0
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36
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Is there a standard procedure to "invert" a multivariable function?
I have a function $P(Q,x)$ ($P$ as a function of variables $Q$ and $x$) and I would instead like to known the function $Q(P,x)$ ($Q$ as a function of variables $P$ and $x$).
These functions are in ...
0
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0
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38
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Finding Sequence of Polynomials Whose Existence is Guaranteed
I'm interested in knowing whether we can find a sequence of polynomials (thanks to Stone-Weierstras) that converges to the Weierstrass function on some random interval (for instance, [-2 , 2]. The ...
- 13
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2
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74
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Nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = (f(x))^2$
Are there any nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the identity $f(f(x)) = (f(x))^2$ $\forall x \in \mathbb{R}$?
1
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1
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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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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 ...
- 3,184
1
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0
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149
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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
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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
113
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Can construct a bijection between R - Q and ( (R - Q) β© [0,1] )?
I've tried to show that:
$$[0,1]\sim([0,1] β©R-Q)$$
I know from this answer :
$$[0,1]\sim R-Q$$
But how to construct a bijection between R-Q and $([0,1]β©R-Q)$ ?
I think the function would be like $f:R-...
- 382
0
votes
1
answer
166
views
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
$$
\begin{aligned}
f\left(f\left(x^{2}-1\right)\right) &=\left(x^{2}-1\right)^{2}-1 ...
- 55
2
votes
1
answer
169
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Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
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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
votes
3
answers
208
views
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
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2
answers
53
views
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
0
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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 $\Bbb ...
user873532
0
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3
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72
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{$\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
votes
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
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2
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312
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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 $...
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