All Questions
Tagged with real-numbers functions
183
questions
7
votes
2
answers
548
views
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
votes
0
answers
30
views
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
views
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
votes
2
answers
70
views
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
votes
1
answer
44
views
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
0
votes
1
answer
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
0
votes
1
answer
108
views
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
0
votes
0
answers
124
views
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
votes
0
answers
47
views
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
0
votes
1
answer
1k
views
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
0
votes
0
answers
37
views
Bijection from $\mathcal{P} (\mathbb{R})$ to the set of functions from $\mathbb{R}$ to $\mathbb{R}$ [duplicate]
I’m 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
answers
1k
views
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
0
votes
0
answers
36
views
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
views
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
votes
2
answers
74
views
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 $f(f(x)) = (f(x))^2$ $\forall x \in \mathbb{R}$?
