Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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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 ...
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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 ...
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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 ...
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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, ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 $$x^2 -7 \lfloor x\rfloor +5=0$$
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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, ...
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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 ...
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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 $$g_2(f(g_1(x))=g_2(f(a_1x+b_1))$$ ...
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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 : ...
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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 $x>0$ ...
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0answers
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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 ...
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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 $f \circ g \neq \textrm{id}_{\mathbb{N}}$. ...
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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 ...
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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 ...
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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, ...
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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 ...
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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?
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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,...
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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?
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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.
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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 ...
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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 ...
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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/\...
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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 ...
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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 $$\int_0^1 xf(x)\,\mathrm dx=\int_c^1 f(x)\,\mathrm dx.$$
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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+\...
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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+\...
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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?
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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
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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 ...
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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?
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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?
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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] \ \ ...
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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 ...
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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
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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 ...
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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?
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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(...
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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}...
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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
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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 $\...
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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\...
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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 ...
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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
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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) $C: \mathscr P (\{1,2,3\}) \to\mathscr P(\{1,2,3\})$ ...
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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)<...

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