Tagged Questions
2
votes
2answers
29 views
Proving the coefficient of Power series is “0” always on given condition.
Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
1
vote
2answers
27 views
Proving a condition on a cont-differentiable function on positive real numbers.
Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$.
I am not getting how we will ...
4
votes
5answers
156 views
Checking whether a polynomial of high degree is bijective or not.
Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false?
I think is when we take ...
1
vote
1answer
49 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
2
votes
1answer
33 views
When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?
I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
0
votes
1answer
40 views
under what conditions is f(A ∪B)=f(A) ∪f(B) and f(A∩B)=f(A)∩f(B)? [duplicate]
Does the function need to be bijective? I know for f(A∩B)=f(A)∩f(B) the function has to be injective, but what about the first equation?
1
vote
1answer
44 views
Proof involving functions.
Consider two functions $f\colon A \to B$ and $g\colon B \to C$. How can I prove the following?
If $f$ and $g$ are one-to-one, then the composition function $g \circ
f$ is one-to-one.
If $f$ and ...
1
vote
6answers
77 views
A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$?
What is the range of the function
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$
1
vote
2answers
38 views
Find a polar representation for a curve.
I have the following curve:
$(x^2 + y^2)^2 - 4x(x^2 + y^2) = 4y^2$ and I have to find its polar representation.
I don't know how. I'd like to get help .. thanks in advance.
8
votes
1answer
148 views
Function mapping challange
For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant
mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$
and $f(k) = f(f(k + 1))$.
This is ...
3
votes
2answers
49 views
Can it be shown that a set X is infinite if and only if there exists some $F:X\to X$ that is an injection but not a surjection?
If the function is not surjective, then at least one element of the codomain has no pre-image. However, because F is a function, every element in the domain is mapped to something in the codomain. ...
2
votes
4answers
59 views
Solve for $x$: $4x = 6~(\mod 5)$
Solve for $x$: $4x = 6(mod~5)$
Here is my solution:
From the definition of modulus, we can write the above as $ \large\frac{4x-6}{5} = \small k$, where $k$ is the remainder resulting from ...
2
votes
1answer
29 views
Functions: Detirmining values a & b
The problem
$f(x)$ and $g(x)$ are defined over the real number set $\mathbb{R}$ as follows:
$$
\begin{split}
g(x) &= 1-x+x^2\\
f(x) &= ax+b
\end{split}
$$
If $g(f(x)) = 9x^2 - 9x + 3$, ...
2
votes
3answers
133 views
Differential calculus - Reviewing and drawing graph
I have missed math class for a few weeks and I'm quite behind with the new stuff learned by the others, so I'm stuck with a problem here. The main problem is, I'm going to have hard time explaining ...
0
votes
2answers
45 views
Where is this function welldefined?
Can this function
$$\left(\frac{3a^2-a}{15+3a}\right)\left(\frac{25-a^2}{9a^2-6a+1}\right)\left(\frac{3}{a}-9\right)$$
be simplified to
$$\frac{(5-a)(1-3a)}{(3a+1)}$$
and if that is true a can't be ...
0
votes
2answers
34 views
Proving bijectivity f: ℚ² → ℚ²
This is what I have to proof bijectivity for:
f: ℚ² → ℚ² : (x,y) ↦ (3x + y, x + 2y)
First I have to proof that the function is injective by doing:
f(x,y) = f(x',y')
And that's where I ...
1
vote
1answer
74 views
Is this claim true that $g\circ h$ is bijection?
Please help me to probe the truth of the following statement.
if $g:Y \to Z$ , $h:X \to Y$ and $g\circ h$ is bijection $\Rightarrow$ $g$ and $h$ are bijection too.
0
votes
4answers
113 views
Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, … $ but is continuous at every other point
(a) Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, ... $ but is continuous at every other point.
(b) Find a function $f:\Bbb R \to \Bbb R$ which is ...
0
votes
1answer
46 views
Calculating the range of $f(x)=\sqrt{2x-6}$.
Please tell me how to calculate the range of $f(x)=\sqrt{2x-6}$?
The solution in my math note says that the range of $f(x)$ is $\lbrace y\in\mathbb{R}:y\ge 0\rbrace$.
As I followed this link and ...
2
votes
1answer
62 views
If $f '(2) = 0$ and $f ''(2) = 4$, what can you say about $f$?
I was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me to ...
2
votes
2answers
70 views
Prove that f(X) is constant.
Now I have seen a lot of answers around here which seem to be good enough.
Problem is, our teacher asked us to prove it his way.
Suppose we know that
$$|u(x)−u(y)|≤(x−y)^2$$
Prove, by adding and ...
5
votes
2answers
62 views
Sets, functions and relations problem
Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes:
Question:
Context:
Let $A$ and $B$ be subsets ...
0
votes
1answer
109 views
Discrete Math functions and sets
Let $A$ and $B$ be subsets of $\Bbb{Z}$, and let $F = \{f : A\to B\}$. Define a relation $R$ on $F$ by: for any $f,g\in F$, $fRg$ if and only if $f - g$ is a
constant function; that is, there is a ...
0
votes
2answers
43 views
Product of Odd Functions
So I'm working on a mini-project for my intro proof writing course and we're given the following that I'm a little hung up on.
Consider $V$ to be a known vector space and functions $f$ and $g$ such ...
0
votes
2answers
85 views
Function and equivalence relations question
Let A and B be subsets of the set Z of all integers, and let F denote the set of all functions
f : A to B. Define a relation R on F by: for any f,g element of F, fRg if and only if f - g is a
...
0
votes
1answer
70 views
Upper and lower bound of $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$
Let define $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$
Please help me prove, that $f$ reaches its lower bound in only one point $x_1$ and reaches its upper bound $x_2$ also in only one ...
0
votes
1answer
30 views
Number of Equivalence Classes
Let $M=\{1,2,\ldots,20\}$ and define a function $f:M\to \mathbf{Z}$ by $f(x)=\min(x,3)$. Define an equivalence relation on M by letting two element $m$ and $n$ be equivalent if $f(m)=f(n)$. 1) How ...
1
vote
2answers
50 views
What does it mean to be proportional to something?
I am asked a physics homework question, but it is really simply a mathematical question I think, dealing with proportional reasoning.
The period of a pendulum is proportional to the square root of ...
0
votes
1answer
98 views
Question about a counterexample related to the mean value theorem for integrals
Let $g(x) = x$ on the interval $[ 1, 3]$. Can you find a function $f (x)$ such that no value between the minimum and maximum of $f (x$) satisfies
$$
\int_{a}^{b}f(x)g(x) dx \,=\, ...
4
votes
0answers
77 views
Bijective map from two injective functions
A bijection $f \colon X \to Y$ is constructed from the injective functions $g\colon X \to Y$ and $h \colon Y \to X$. Suppose $X = Y = \mathbb N$. We let $g\colon X \to Y$ and $h \colon Y \to X$ be ...
6
votes
1answer
137 views
Exercise Functional Analysis
Let $\mathcal{F}$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$.
Consider an operator $\mathcal{O}: \mathcal{F} \rightarrow \mathcal{F}$ such that:
$\mathcal{O}( f_1 + f_2) = ...
0
votes
1answer
29 views
Are those functions surjective, how would you prove it
Consider the functions $\text{add}:\mathbb{A}\to \mathbb{Z}$, such that $\mathbb{A}$ is a finite subset of $\mathbb{Z}$, and $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$
...
3
votes
1answer
28 views
Beginning Proof Question concerning Functions
So my class has been given the task to find functions $f$ and $g$,both from R to R such that:
$f+g$ is differentiable and either $f'(0)$ dne, $g'(0)$ dne or both.
I'm starting to believe, or at least ...
0
votes
1answer
52 views
Writing a formula for a spread sheet that can solve for a given value
For instance if given values are $f(1.1) = 1$, $f(1.9) = 2$, $f(2.7) = 3$, $f(3.5) = 4$, $f(4.3) = 5$ and ever increasing by $0.8$ for each number, if I were to enter a value of $1.5$ the answer is ...
2
votes
1answer
33 views
Finding domain of a function
Finding the domain of the function
$$f(x)=\frac{3\sqrt{x}}{x^2-5x-14}$$
My working out
First I factorise the denominator
$$f(x)=\frac{3\sqrt{x}}{(x-7)(x+2)}$$
Therefore the Domain is bigger than ...
1
vote
3answers
75 views
If $\log(a) -\log(b) \gt \log(c) -\log(d)$, is $(a-b) \gt (c-d)$?
If $\log(a) - \log(b) \gt \log(c) - \log(d)$, must it always follow that $(a-b) \gt (c-d)$?
I'm looking over someone else's work, and they inexplicably took the logarithms of the numbers and then ...
0
votes
0answers
32 views
I wanna prove if the composite are equal to each other
Given $f : \{0,1\}^n \to \{0,1\}^n$, define $f': \{0,1\}^{2n} \to \{0,1\}^{2n}$ as follows: for $x, r \in \{0,1\}^n$ define $f'(x \circ r) := f(x) \circ r$ (where $\circ$ denotes concatenation). Prove ...
0
votes
2answers
99 views
Prove that $ f(0) \neq 0 $.
Let $ f: \mathbb{R} \to \mathbb{R} $ be a non-constant function such that $ f(a + b) = f(a) \times f(b) $ for all real numbers $ a $ and $ b $.
a) Prove that $ f(0) \neq 0 $. (Hint: Otherwise, $ f(x) ...
1
vote
2answers
50 views
Showing that a modifying function which is continuous at 0 is uniformly continuous
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
1
vote
1answer
45 views
Showing that a function is a modifying function (how to prove subadditivity)
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
0
votes
1answer
53 views
Determining If A Relation Is A Function
I am given the simple relation $f(x)=\sqrt{x}$, where $f$ maps $R \rightarrow R$, and I am suppose to determine whether or not it is a function.
I figured that it was a function, because in the ...
6
votes
2answers
114 views
equivalence of norms
I would like a little help here:
I have two defined norms over $C^{1}([0,1])$ :
$\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$
$\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$
I already ...
4
votes
2answers
137 views
Prove that function is bijective
Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$.
...
0
votes
5answers
45 views
Finding The Domain Of A Function Of Two Variables
The function is $f(x,y) = \sqrt{4-x^2-y^2}$
I know that the only allowable $x$ and $y$ are those that yield either zero under the square root symbol, or some positive number. With that in mind,
...
4
votes
1answer
99 views
Using tan(x), show that open interval is diffeomorphic with the real line
This is exercise 1.3 from An Introduction to Manifolds by Loring W. Tu
I want to show that $(a,b) \subset \mathbb{R}$ is diffeomorphic with the real line.
Stategy:
1.) I want to ...
0
votes
1answer
57 views
How can I show that f is a diffeomorphism?
Let $G=\{(x,y,z)\in \mathbb{R^3}\mid x^2+y^2-z^2+1=0; z>0\} $ and $D=\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 <1\}$.
Let $f\colon G \rightarrow D $ be a mapping such that $$f(x,y,z)=\left( \frac x ...
2
votes
2answers
54 views
questions on a continuous, injective, surjective
Let $f: X\rightarrow Y$ be a continuous, injective, surjective.
Question 1, if $f$ is open or closed, then does $f^{-1}$ continuous?
Question 2, if $f$ is open or closed, then does $f^{-1}$ open or ...
0
votes
0answers
32 views
Expressing functions using Karnaugh map [duplicate]
Using the Karnaugh map, express the following function:
$F(0, 1, 4, 5, 8, 10, 11, 12, 13, 15)$
would this be the answer
I'm a little confuse
($b_1=0$ and $b_0=0$) or ($b_3=0$ and $b_1=0$) or ...
0
votes
2answers
134 views
Addition function injective?
I am just curious to know if addition of two numbers an injective function?
Lets say $\operatorname{Sum}(a,b) = a + b$
Now is the $\operatorname{Sum}$ function an injective functions?
0
votes
3answers
71 views
The square of a measurable function is measurable
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a measurable function.
I want to show that $f^2:x\mapsto (f(x))^2$ is measurable.
Apparently it can be shown using the facts that the sum of two ...
