All Questions
Tagged with real-numbers functions
183
questions
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43
views
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
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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
votes
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
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
0
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4
answers
67
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Rudin PMA 4.31 - does the elements of $E$ have to be ordered (smallest to the biggest)?
Here is a reformulation of Rudin PMA $4.31$ remark:
Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ ...
- 1,231
2
votes
2
answers
59
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Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
7
votes
2
answers
157
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Find all real functions that satisfy: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ $\in$ $\mathbb{R}$
I am trying to find all real functions that satisfy the property: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ real numbers. I tried to substitute $x$ and $y$ with $0$ but end up with nothing, then I ...
- 301
1
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1
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98
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Functional square root of a function $F([a,b])=[1βbr+a,1βbr+a+b]$.
Suppose we have a function $F\colon \mathbb{R}^2 \to \mathbb{R}^2$ over a numeric pair (pair of real numbers) $[a, b]$ such that:
$$F([a,b]) = [1-br+a, 1-br+a+b]$$
for some $r\in\mathbb{R}$.
find a ...
- 67
2
votes
2
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122
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Formula for the $n$-th term of the sequence $1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$, where $f_n := \frac{1}{n} (f_{2n} - f_n)$
I'm struggling with this sequence.
$$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
votes
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
0
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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
votes
1
answer
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
2
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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 $f(x,y) = (f_1(x,y),f_2(x,y))$.
...
- 177
-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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0
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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 ...
0
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0
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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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1k
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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 ${[0,1]}^{R}$...
- 427
6
votes
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
votes
0
answers
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
0
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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 $f(f(x)) = (f(x))^2$ $\forall x \in \mathbb{R}$?
