All Questions
66
questions
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
vote
2
answers
119
views
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
0
votes
1
answer
46
views
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
views
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
answer
26
views
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
2
votes
2
answers
59
views
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
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
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
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 ...
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
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}$?
6
votes
2
answers
367
views
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
0
votes
1
answer
41
views
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