Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
25,037
questions
-1
votes
0answers
9 views
To prove that finite subset of powerset of infinite number of symbol is countable. Prove that {S ∈ P({b} ∗ ) | S is finite } is countable.
Let b be a symbol. Prove that {S ∈ P({b}
∗
) | S is finite } is countable.
I'm not sure how to represent b and hence cannot proceed.
I know I'm supposed to make a bijection to prove countability
Can ...
0
votes
1answer
15 views
Find the max $\{\sqrt{n^3}\lg n, \sqrt[3]{n^4}\lg^5n \}$
*Note: the logs are with base 2 (computer scince question).
Let
$$
f(n) = \max\{\sqrt{n^3}\lg n, \sqrt[3]{n^4}\lg^5n \} = \max\{f_1(n),f_2(n)\}
$$
I want to find which function is the max between ...
1
vote
2answers
32 views
I need help with a proof. Prove that the set of bijections $f:[0,1]\to[0,1]$ , such that $f=f^{-1}$ is uncountable.
Prove that the set of bijections $f:[0,1]\to[0,1]$ such that $f=f^{-1}$ is uncountable.
That's all that I'm given in the text of the assignment, it's my first time doing proofs so I don't even know ...
1
vote
1answer
13 views
What is the y=mx+b equation of a train bowing at 392 km after 2.8 hours, leaving the station at x=0?
A train leaves the station at time x=0. Traveling at a constant speed, the train travels 392 km in 2.8 hours. Round to the nearest 10 km and the nearest whole hour. Then represent the distance, y, ...
0
votes
0answers
29 views
Domain amd Range of Composite functions
Let $f(x)=\begin{cases}2x+a,&x\geqslant -1\\bx^2+3, &x\lt -1\end{cases} \;\\g(x)=\begin{cases}x+4, &0\leqslant x\leqslant 4\\3x-2,& -2\lt x\lt 0\end{cases}$
Find the range of $a$ and $...
0
votes
3answers
26 views
Global Maxima of a function
Let $x=e^{-3t}(3 \cos(4t)+\frac{9}{4} \sin(4t)), t\geq 0$. Prove that $\lvert x \rvert \leq 3$.
Now, using a graphical approach, the above result is clear. However, is there a way to prove the above ...
0
votes
1answer
22 views
Is it correct to write the range of the sigmoid function as [0, 1]?
The sigmoid function is defined as follows:
$$\sigma (z) = \frac{1}{1+e^{-z}}.$$
Hence, $\sigma (z) = 0$ when $z$ is minus infinity, and $\sigma (z) = 1$ when $z$ is infinity.
Why is the range of ...
0
votes
2answers
20 views
Finding example of a function of a set of all integers to the set of positive integers
OK, so I have run into this weird question about recurrence relations that I cannot complete by myself (first year comp. sci. student and first discrete math class, studying by myself due to ...
2
votes
2answers
32 views
What is the sum of the squares of the roots of the equation? [closed]
What is the sum of the squares of the roots of the equation $$x^2 -7 \lfloor x\rfloor +5=0$$
0
votes
0answers
18 views
Generalized quadratic loss learning
I'm studying a binary classification task with an objective function, derived from SVM, defined so:
$\sum_{i=1}^{l} \sum_{j=1}^{l} \vec{\xi}' S \vec{\xi}$
with:
$y_i (f(\vec{x}_i)) >= 1 - \xi_i, ...
0
votes
1answer
12 views
Question about functions?
Suppose you have two functions $f: X \rightarrow \mathbb{R}$ and $g : X \rightarrow \mathbb{R}$. I was able to show that $(f+g)(x) = (g+f)(x) \forall x \in X$, and as far as I can tell both maps have ...
-1
votes
0answers
32 views
Find two linear functions $g_1$ and $g_2$ such that $(g_2\circ f\circ g_1)(x)=x^2$
For $f(x)=3x^2-2x-7$, find two linear functions $g_1$ and $g_2$ such that $$(g_2\circ f\circ g_1)(x)=x^2$$
First of all I let $g_1=a_1x+b_1$ and $g_2=a_2x+b_2$
then $$g_2(f(g_1(x))=g_2(f(a_1x+b_1))$$
...
0
votes
4answers
40 views
Is square root a function?
I just began the topic of functions in my Mathematics textbook and in the first paragraph itself, two conditions about a relation being a function were mentioned.
They were :
...
3
votes
1answer
58 views
Find all functions $f:\mathbb{R} \to [0, \infty)$such that $f(x^2 + y^2)=f(x^2 - y^2)+ f(2xy)$.
Question -
Find all functions $f:\mathbb{R} \to [0, \infty)$ such that
$$f(x^2 + y^2)=f(x^2 - y^2)+ f(2xy)$$ for all $x,y\in\mathbb{R}$.
My try -
$f(0)=0$ and $f(x)=f(-x)$ for all $x>0$ ...
1
vote
0answers
13 views
Finding the maximum value of a function found from a differential equation
I solved a differential equation and got the function $Q=\frac{t}{2}e^{-\frac{t}{20}}$. I am being asked to then find the point in which the value is at its maximum. One solution approach I can think ...
-1
votes
1answer
14 views
composition of functions - one-way inverse [closed]
Construct functions $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow \mathbb{N}$ such that $g \circ f = \textrm{id}_{\mathbb{N}}$ but $f \circ g \neq \textrm{id}_{\mathbb{N}}$. ...
1
vote
1answer
62 views
Increasing Function that is discontinuous on A
Given any set $A \in \mathbb R$ where $m(A)=0$, I need to construct a function that has derivative of $+\infty$ on $A$ and the function needs to bounded. At this point I want to make it monotonically ...
0
votes
1answer
60 views
Can there be a continuous function $f:\Bbb R→\Bbb R$ where $f(x)\in\Bbb Q$ $\Leftarrow\Rightarrow$ $f(x+1)\notin\Bbb Q$ [closed]
Prove that there exists at least one continuous function with the specified property, or proof that no such function can exist.
The property is equivalent so if $f(x)\in\Bbb Q$ then $f(x+1)\notin\Bbb ...
0
votes
0answers
8 views
Real anaylisis. Continuity of a product [duplicate]
Let $X$ $\subset$ $\mathbb{R}$ and $f,g:X\rightarrow\mathbb{R}$ two bounded funtions. Is the prouduct $fg$ a uniformly continuos function? Prove or give a counterexample.
I would say it is false, ...
0
votes
1answer
12 views
Is there a method to find the result of the rotation of a function in $\mathbb{R^2}$ to a shape in $\mathbb{R^3}$
For example, if i rotate the function $f(x) = \sqrt{1-x^2}$ over the $x$ axis, i would get the shape $x^2 +y^2 +z^2 = 1$.
And if i rotate the function $g(x) = \sqrt{x} $ over the $x$ axis, i would ...
-1
votes
0answers
15 views
how many function are there on set with n elements [closed]
for a set containing n elements
How many functions are there on this set with n elements?
and how many of them are bijective function?
0
votes
3answers
22 views
Function different from original by differentiating it and integrating it again. [duplicate]
Let a function be defined $f(x)=sin^4x+cos^4x,$ $x\in R$.Rewriting this,
$$f(x)=(sin^2x+cos^2x)^2-2sin^2xcos^2x$$$$\implies f(x)=1-\frac{sin^22x}{2}$$$$\implies \frac{1}{2}\leq f(x)\leq1 $$
However,...
0
votes
0answers
41 views
What does the “max” do in a function $\max_{\alpha \geq 0} x^{3} + \alpha \left(4 - x \right)$?
In the function $\max_{\alpha \geq 0} x^{3} + \alpha \left(4 - x \right)$, what does the max do in this function?
1
vote
1answer
5 views
Existence of continuous surjection from open unit square to closed unit square
Does there exists a continuous surjection from (0,1)×(0,1) to [0,1]×[0,1]?
I have found such a function from (0,1) to [0,1]. I just want to generalize it.
0
votes
0answers
15 views
Properties and Uses of the Mellin Transform [closed]
What is the Mellin transform really? I've understood how you complete a Mellin transform, but what uses has it? For example, does it share any properties with the original function? Is it nonzero for ...
1
vote
1answer
24 views
Can we integrate an inequality under specific conditions?
While studying inequations I noticed some differences in how equation and inequations are solved.
Among many things we cannot differentiate a inequation like a equation(as a function being greater ...
0
votes
2answers
39 views
Asymptotic Expansions
Can someone explain this to me, please
(a) $\sin (1/\varepsilon) = O(1)$ as $\varepsilon \to 0$,
but it is not true that $1 = O(\sin (1/\varepsilon))$ as $\varepsilon \to 0$ because $\sin (1/\...
1
vote
1answer
39 views
How to use induction on $p+q$ in functional equations
Question -
Find all $f : \Bbb Q^+\to\Bbb Q^+$
such that
a) $f(x)+f(1/x)=1$
b) $f(f(x))=f(x+1)/f(x)$.
My try -
By checking some values I get
$f(1)=1/2$
$f(2)=1/3$
.....
and using induction I ...
0
votes
1answer
27 views
Continuous functions integral proof
If $f$ is a continuous function $f\colon [0,1]\to \Bbb R$ then show that there exists a point $c \in (0,1)$ such that $$\int_0^1 xf(x)\,\mathrm dx=\int_c^1 f(x)\,\mathrm dx.$$
-1
votes
1answer
31 views
I need help with this simple problem in little-o notation [duplicate]
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\...
-2
votes
1answer
40 views
Could anyone explain to me how did we get this result? This is simple example but I'm struggling with little o notation in general. [closed]
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\...
1
vote
1answer
37 views
Is there any other function similiar to Weierstrass function, that is continuous at every point but non-differentiable at any point on a compact?
I’m studying functional series and right now we’re dealing with Weierstrass function. I was wondering if it’s the only known function with this property?
0
votes
0answers
16 views
Finding f(g(h(x)))=g(h(f(x)))
This may be a silly question but I was curious if it was possible to find a series of algorithms that would satisfy the property of f(g(h(x)))=g(h(f(x)))
For context I’m attempting to determine a ...
2
votes
1answer
29 views
Proving The existence of Set Y from non-empty Set X with an equivalence relation defined on X.
Let $X$ be a nonempty set and let $\sim$ be an equivalence relation defined on $X$.
Prove that there exist a set $Y$ and a function $f \colon X \to Y$, such that for all $a, b \in X$:
$$a \sim b \iff ...
0
votes
1answer
20 views
Function defining $nCr$
While playing with my calculator I found that surprisingly, it is giving values for fractional values of $n$ and $r$
How is that possible?
3
votes
1answer
59 views
Assumption of polynomial
Let f(x) satisfy the equation
$f^2(x)=1+xf(x+1)$ and $x+1 <2f(x)<4(x+1)$ for all x > 1. Find f(x).
I tried with assuming polynomial, but of no help. Is assuming polynomial correct?
2
votes
1answer
7 views
Find the average of the Fat cantor set based on its distribution
Suppose we have $P:A\to[0,1]$, where $A$ is the fat cantor set denoted as $C$.
We produce $C$ by removing $1/4$ of $[0,1]$ around mid-point $1/2$
$$C_{1}=[0,3/8]\cup [5/8,1]$$
$$C_{1,1}=[0,3/8] \ \ ...
0
votes
2answers
29 views
What is a function that converts any number to a one, while keeping the sign intact?
Given a number $x$, I would like an $f(x)$ whose value always equals 1, but keeps the sign of $x$ intact. So if $x = 5$, then $f(x) = 1$. If $x = -13$, then $f(x) = -1$ and $f(0) = 0$
What is the ...
0
votes
0answers
15 views
Explaining in detail the contents of this question [closed]
Can someone explain each function in this example. I find it all very confusing.
Let $A = \{1,2,3,4\}$. Let F be the set of all functions from $A$ to $A$. Recall that $I_A \in F$ is the identity ...
2
votes
4answers
64 views
Is $\lim_{n\to \infty}f(x_n)=f(\lim_{n\to \infty}x_n)$ always true?
I was studying about sequences of numbers and their limits. My book states the standard rules for algebra of limits involving sums, differences, products and quotients of convergent sequences.
But ...
0
votes
0answers
21 views
Composition, bijectivity [closed]
Suppose $f\colon A\to B$ is a composition $f=h\circ g$, where $h\colon C\to B$ is injective and $g\colon A\to C$ is surjective. Can we say that $\tilde{h}\colon C\to \textrm{im}f$ is bijective?
0
votes
2answers
59 views
Find all $f$ such that $f(x+y)=f(x)f(y)f(xy)$ for all $x,y \in \mathbb{R}$
Question -
Find all $f : \mathbb{R} \to\mathbb{R}$ such that
$f(x+y)=f(x)f(y)f(xy)$ for all $x,y \in \mathbb{R}$.
My try -
Putting $x=y=0$ I get three cases $f(0)=0$,$f(0)=1$,$f(0)=-1$
For $f(...
0
votes
0answers
13 views
Summation with terms involving exponent of exponent
If $f(x)$ $=$ $\sum_{n=0}^\infty$ $\frac{x^{2^{n}}}{1-x^{2^{n+1}}}$ , $x$ $\epsilon$ $(0,1)$
Find the value of $f^{-1}({\frac{8}{5f(\frac{3}{8})}})$
My approach till now:
$-f(x)$ $=$ $\sum_{n=0}...
0
votes
1answer
12 views
Simple evaluation of function
Let $F(x, y) = (x^{-\alpha} + y^{-\alpha} - 1)^{-1/\alpha}$ for some $\alpha > 0$. Show that
$$F(0, y) = 0$$
According to my textbook this is trivial, but I really don't understand. How do I ...
2
votes
1answer
61 views
Surjective vector field on $\mathbb{R}^n$
Let $V: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous with the property
$$\frac{\langle V(x), \, x\rangle}{|x|} \, \to \infty \quad \text{as} \quad |x| \to \infty \qquad \qquad (1)$$
where $\...
-1
votes
0answers
36 views
Interesting functional equation [closed]
Find all infinitely differentiable functions of two real variables that satisfy the functional equations
$f\left(x, y\right)=f\left(y, x\right)$
$f\left(x, y\right)+f\left(y, z\right)+f\left(z, x\...
0
votes
1answer
52 views
Olympiad Functional equation
Question -
Find all functions $f:(0, \infty) \to (0, \infty)$ such that
a) $f(x) \in (1, \infty) \forall x \in (0,1)$
b) $f(xf(y))=yf(x) \forall x,y \in (0, \infty) $
(All $\infty$ are positive ...
-1
votes
1answer
48 views
what is the figure for $f(x)=|\sqrt{x}|$
what is the figure of $f(x)=|\sqrt{x}|$ ?
when I run y=abs(sqrt(x)) to matalab, i take this but I believe that is not the correct answer. Please give me a help:
enter image description here
-1
votes
0answers
34 views
m: Z5 -Z5 defined by m(x) = 3x mod 5.
Make a table for the following functions:
a)
$$
m: \mathbb{Z}_5 \to \mathbb{Z}_5 \quad \text{defined by}~\quad m(x) = 3x \mod{5}
$$
b) $C: \mathscr P (\{1,2,3\}) \to\mathscr P(\{1,2,3\})$ ...
0
votes
0answers
23 views
Showing that A is continous at 0.
This is question
And this is answer
The first verification step is pretty intutive for me since it uses standard way to verify $A(\dfrac{1}{n})\to1=A(0)$. My problem is The statement $-1/(N+1)<...
