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
120
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 ...
- 3,019
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
551
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$...
- 763
0
votes
1
answer
109
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
$$
for all $ x \in \mathbb R $, and $ h ( 0 ) = 0 $. What would be an as mild as ...
- 12.1k
0
votes
1
answer
41
views
What is the correct inverse function for $f(x) = x^2$. Question about terminology.
Is the inverse function $g_1(x)=\sqrt{x}$ or is it $g_2(x)=-\sqrt{x}$. With what terminology can we describe both of these functions?
- 2,527
2
votes
1
answer
171
views
Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
0
votes
3
answers
72
views
{$\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
3
votes
1
answer
81
views
Suppose $\sum_{n\ge 1} |a_n| = A<\infty.$ Under what conditions is $\sum_{n\ge 1} \epsilon_n a_n = [-A,A]$, for $\epsilon_n \in \{-1,1\}$?
Consider the space of sequences:
$$
\mathcal{E} = \{\{\epsilon_n\}_{n= 1}^{\infty}: \epsilon_n = \pm 1\}
$$
This can be considered a "random choice of sign" in the probabilistic context, for ...
- 8,369
0
votes
3
answers
94
views
Is it true that there is a bijection $[0, 1) \to \mathbb{R}$?
Is there is a bijection from $[0,1)$ to $\mathbb{R}$?
I thought of an instance, $$\frac{\sqrt{x(1-x)}}{x-1}.$$
- 127
0
votes
1
answer
42
views
Explain a confusing bound for the integral of a decreasing function.
I am reading a solution of an exercise. In the solution, it says the following:
Consider $g(x,t):=\frac{x}{(1+tx^{2})t^{\alpha}}$, where $x\in (0,\infty)$, $t=1,2,3,\cdots$ and $\alpha>\frac{1}{2}$...
- 5,930
-2
votes
2
answers
50
views
Let $a$ be a real number such that $a > 0$. Show that the function $f : [a, +\infty) \to \mathbb R, f(x) = \tfrac{1}{x}$ is uniformly continuous.
Let $a$ be a real number such that $a > 0$. Show that the function $$f : [a, +\infty) \to \mathbb R, f(x) = \dfrac{1}{x}$$
is uniformly continuous.
- 1
0
votes
1
answer
106
views
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have $|f(x) − f(y)| < |x − y|$
Let $a$ and $b$ two real numbers such that $a < b$ and $f : [a, b] \to [a, b]$.
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have
$|f(x) − f(y)| < |x − y|$. Show that ...
- 27
3
votes
2
answers
86
views
If following actions allowed, Find $F(2002,2020,2200)?$
If following actions allowed,Find $F(2002,2020,2200)?$
$$ F(x+t,y+t,z+t)=t+F(x,y,z);$$
$$ F(xt,yt,zt)=tF(x,y,z);$$
$$ F(x,y,z)=F(y,x,z)=F(x,z,y)$$
where x,y,z,t are real numbers.
My attempt:
$F(0,0,0)...
- 59
0
votes
2
answers
89
views
Is there a name for a real-valued function whose input is also real?
I'm trying to write a sentence about a function $f:\mathbb{R}\to \mathbb{R}$, and I want to refer to it as real valued or as a scalar function, or some similar term, but I want that term to also ...
- 1,361
9
votes
3
answers
153
views
How different can $f(g(x))$ and $g(f(x))$ be?
Given $f,g: \mathbb{R} \rightarrow \mathbb{R}$, how "different" can $f(g(x))$ and $g(f(x))$ be?
By "how different" I mean:
Given two real-valued functions $a,b$ do there exist two real-valued ...
- 2,471
-1
votes
1
answer
132
views
Is a function $f(x)=\ln({x^2-1})$ even and symmetric
We have a function:
$$
f(x)=\ln(x^2-1)
$$
The function is symmetric because: $D_f=(-\infty,-1) \ \cup\ (1,\infty)$
I understand this as if we would multipy this by $-1$ we would get the same $D_f$
...
- 2,527
2
votes
2
answers
208
views
Why is this function continuous on $\mathbb R$?
Let $f:\Bbb{R}\to\mathbb R$ be a function with $f(0) = 1$ and $f(x+y) \le f(x)f(y)$ for all $x, y \in \Bbb{R}$. Prove that if $f$ is continuous at $0$, then $f$ is continuous on $\Bbb{R}$?
THOUGHTS:
...
- 160
1
vote
2
answers
231
views
Set of constant functions are uncountable.
Let $F=$ $\{$ $f: [0,1] \rightarrow \mathbb{R}$ $:$ $f$ is constant$ \} $. I must show that $F$ is uncountable.
Note, that for any $f \in F$, and any $c\in \mathbb{R}$, I will denote the constant ...
user643073
2
votes
0
answers
72
views
Proving integral of a continuous function is continuous
Let $U \subset \mathbb{R}^N$ be an open set, let $f : U \times [a, b] \to \mathbb{R}$ be a continuous function. Consider the function $$g(x):= \int_a^b f(x,y) \,dy$$
with $x \in U$.
i) Prove ...
- 1,913
2
votes
2
answers
461
views
Strictly increasing bounded function of class $C^1$
Let $f:\mathbb{R} \to \mathbb{R}$ be a strictly increasing bounded function of class $C^1$. Prove that there exists a sequence $\{x_n\}_n$ of real numbers such that $x_n\to\infty$ and $\lim_{n \to \...
- 1,913