All Questions
Tagged with real-numbers functions
183
questions
1
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0
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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
views
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
vote
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
answers
677
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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
0
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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 ...
- 2,947
0
votes
1
answer
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
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
- 33
0
votes
4
answers
67
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Rudin PMA 4.31 - does the elements of $E$ have to be ordered (smallest to the biggest)?
Here is a reformulation of Rudin PMA $4.31$ remark:
Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ ...
- 1,221
2
votes
2
answers
59
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Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
7
votes
2
answers
157
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Find all real functions that satisfy: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ $\in$ $\mathbb{R}$
I am trying to find all real functions that satisfy the property: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ real numbers. I tried to substitute $x$ and $y$ with $0$ but end up with nothing, then I ...
- 301
1
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1
answer
98
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Functional square root of a function $F([a,b])=[1βbr+a,1βbr+a+b]$.
Suppose we have a function $F\colon \mathbb{R}^2 \to \mathbb{R}^2$ over a numeric pair (pair of real numbers) $[a, b]$ such that:
$$F([a,b]) = [1-br+a, 1-br+a+b]$$
for some $r\in\mathbb{R}$.
find a ...
- 67
2
votes
2
answers
122
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Formula for the $n$-th term of the sequence $1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$, where $f_n := \frac{1}{n} (f_{2n} - f_n)$
I'm struggling with this sequence.
$$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