Elementary questions about functions, notation, properties, and operations such as function composition.

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0
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1answer
19 views

is this the way to evaluate these functions? if not, how can I evaluate?

Given that $$ f(x)=1-x, g(x)=x^2-3, h(x)=\frac2x $$ 1) Find $$f\,o\,g$$ My answer 1) $$f\,o\,g(x)=f(g(x))=f(x^2)=3x^2-1$$ 2) Find $$f\,o\,h$$ My answer 2) $$f\,o\,h = f(h(x))= 2/1-x-3 $$ 3) ...
0
votes
4answers
64 views

Limit with the sinus squared function

In my assignment I have to determine if the following limit is true or false: $$\lim_ {x \to 0} \frac {\sin^2(2x)}{\sin(2x^2)}=1$$ I think that it's false. Here is my calculation, and I wanted to ...
4
votes
1answer
43 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...
0
votes
0answers
43 views

Looking for a function which can serve as an upper bound to a count of the the pairs (x)(x+2) that have a given least prime factor?

Let $p \ge 7$ be a prime. Let $z > p$ also be a prime. Let $f_p(z)$ be the number of elements $x$ such that $z \le x < z^2$ and the least prime factor of $x(x+2) = p$ I am trying to find ...
0
votes
0answers
20 views

Help with a proof involving integration of difference of functions

Let $f(x,a)$ and $g(x,a)$ be two continuous functions from $[0,1]\times \mathbb{R}^+ \to \mathbb{R}$. $g(x,a)$ is decreasing and convex in $x$. $f(x,a)$ is decreasing in $x$ and increasing in $a$. ...
0
votes
0answers
14 views

A function to relate all its variables

I'm dealing with a function looks like $z=f(x,y_1,y_2,y_3)$; where $x$ and $y$ are independent variables. Basically what I did so far is: $z=f(x,y_1)$, where $z= \text{energy}$, $x = \text{distance}$ ...
3
votes
1answer
28 views

Is this is the right way to do these one-to-one functions, finding their inverse, if not, how to do it?

Question 1) $f(x) = 1-x$ My answer (1): $f(x) = 1-x$, $y = 1 - x$, $y + 1 = x$, $x = y + 1$, $f$ of inverse $f(y) = y + 1$ Question 2) : $f(x) = \dfrac{2x}{x-1}$ My answer 2) : $f(x) = ...
1
vote
0answers
25 views

How find set $f(\Bbb R^3)$ for $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$?

Let $f:\Bbb R^3 \to \Bbb C$ such that $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$. How can one find the set $f(\Bbb R^3)$?
0
votes
2answers
48 views

Prove/Disprove $f(x)=e^{x}$ is Injective and Surjective

Dr. Pinter's "A Book of Abstract Algebra" presents the exercise: Prove whether each of the functions is or is not (a) injective and (b) surjective. $$f \implies \mathbb{R} \to (0, \infty)$$, ...
1
vote
2answers
46 views

Trying to find a function such that $\lim_{x\to\infty} f(x)=0$ but $\lim_{x\to\infty} f'(x) \ \text{does not exist}$.

I am trying to find a function, which will have these two properties and I cannot really succeed in it. $\lim_{x\to\infty} f(x)=0$ $\lim_{x\to\infty} f'(x) \ \text{does not exist}$ Has ...
0
votes
0answers
42 views

Is there such thing as a Co-theta?

I plan to make a notation for trigonometry called co$\theta$ An example is that co$\theta$ is equal to 45 and $\theta$ is equal to 45. Co$\theta$+$\theta$=90. In this equation, ...
4
votes
1answer
21 views

“More” than nowhere locally bounded function

I describe here a function $g$ (based on Thomae's function) that is nowhere locally bounded. In particular the image of any interval $(a,b)$ under $g$ is an unbounded segment of the integers ...
6
votes
2answers
86 views

$f(x)=x^{3}+1$ - Injective and Surjective?

Dr. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove/disprove whether the following function $f$ (with inputs/outputs of real numbers) is injective and/or surjective: ...
0
votes
2answers
30 views

Real (Valued) Functions in German

I just realized something that was left unnoticed by me for many years. Apparently, among German speakers reelle Funktion (literary also translated word by word as "real functions") has both domain ...
1
vote
1answer
42 views

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$ Find $f([-2,1])$= $[-3,1]$ Find $f^{-1}([-2,1])$= $[-1,1]$ I am not wonderful at these types of problems and I seem to make ...
4
votes
4answers
62 views

$f(x)=3x+4$ - Injective and Surjective?

As a follow-up to Understanding why $f(x)=2x$ is injective, I'm working on proving/disproving that $$f(x)=3x+4,$$ where inputs/outputs live on real numbers, is injective and surjective. Supposing ...
2
votes
3answers
53 views

Let $f:U \to V$ be a bijective holomorphic function. Show that inverse of $f$ is also holomorphic.

Suppose $U$ and $V$ be domains(i.e., open and connected) in $ \mathbb C$.Let $f\colon U \to V$ be a bijective holomorphic function. Show that the inverse of $f$ is also holomorphic. By Open Mapping ...
2
votes
4answers
63 views

Understanding why $f(x)=2x$ is injective

Dr. Charles Pinter's "A Book of Abstract Algebra" shows a proof of why a function '$f$' is injective and surjective. Given $$f(x)=2x,$$ we claim $f$ is injective. Proof. Suppose ...
0
votes
1answer
23 views

Interpreting a condition about CDF

Let F(X) be a strictly increasing CDF which admits a positive density f(x). Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex? If not, what ...
1
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0answers
40 views

Finding probabilities of a continuous random variable

I have the following continuous random variable density function: $$ f(x) = \begin{cases} \frac14 & if\,0\le x<1 \\ \frac12 & if\,1\le x<2 \\ a & if\,2\le x<4 \\ 0 & ...
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votes
2answers
34 views

Using functions to solve functions [closed]

$1.$ Use the function $g(x)=8x-7$ to find the function $g^{-1}(x)$ $2.$ Use the function $g(x)=4x-3$ to find the function $g^{-1}(x)$
1
vote
2answers
39 views

How to find period of functions involving greatest integer function [.]

How to find period of functions involving greatest integer function [.] 1) $[x+1/2]+[x-1/2]+2[-x]$ 2) $[x]+[x+1/3]+[x+2/3]-3x+10$ Hints will suffice.I'm not being able to approach these ...
-1
votes
6answers
81 views

$f(x^2+2x+4)=3x^2+6x+7$, find $f^{-1}(4)$. [closed]

$f(x^2+2x+4)=3x^2+6x+7$, find $f^{-1}(4)$.
1
vote
1answer
32 views

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$. I tried this problem two ways, substituting $x=1,-1$ in $f(x)$ to find the remainder, and by long ...
1
vote
2answers
50 views

Let $n$ be an odd natural number , to find a continuous real valued function on $\mathbb R$ which takes every value exactly $n$ times

Let $n$ be an odd natural number . We know $\mathbb R = \cup_{k \in \mathbb Z} [nk\pi , n(k+1)\pi]$ . So for every $k \in \mathbb Z$ , define $h(x):=2k+1-(-1)^k \cos x , \forall x \in [nk\pi , ...
0
votes
1answer
28 views

How does differentiability affect the extremum of a function?

I have this function $$f(x)= \begin{cases} (x+1)^3 & -2< x\le-1\\ x^{2/3}-1 &-1<x\le1\\ -(x-1)^2 &1<x<2 \end{cases}$$ I'm supposed to find the total number of maxima and ...
0
votes
4answers
48 views

Poker Chips function [closed]

Suppose a poker player has a stack of 33 green poker chips and another stack of 33 white poker chips he then sticks these two stacks next to each other and then mixes the stacks together in an ...
9
votes
1answer
81 views

Interesting properties of the function $(a,b)\mapsto a/(a-b)$

Consider the extremely simple function $$f(a,b)=\frac a{a-b}.$$ This gives the coordinate where the line through $(0,a)$ and $(1,b)$ meets the $x$-axis. I noticed that the function $f$ has some ...
0
votes
2answers
21 views

Define $f : X \rightarrow Y$ by $f(x) = y_0$ for every $x \in X$. Then $f$ is $\mathfrak T_1 - \mathfrak T_2$ continuous.

Suppose that $(X, \mathfrak T_1)$ and $(Y, \mathfrak T_2)$ are topological spaces and suppose $y_0 \in Y$. Define $f : X \rightarrow Y$ by $f(x) = y_0$ for every $x \in X$. Then $f$ is $\mathfrak ...
0
votes
1answer
19 views

Is $f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)}$ quasiconvex?

Is the function \begin{equation} f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)} \end{equation} where $0 \leq \beta \leq \alpha$ and $0 \leq \gamma\leq \alpha$ quasiconvex? $x$ can be assumed to ...
1
vote
1answer
25 views

Measuring the “flatness” of a function

In some work I am doing, for a function $f$, I want to measure the average difference between two function values $|f(x_1) - f(x_2)|$ over the entire data distribution, $\int_X \int_X |f(x_1) - ...
-1
votes
0answers
19 views

Randomly incrementing function [closed]

I'm trying to make a fake download count. It should increment randomly over time. Some download-count-like patterns would be nice. The download count should be ...
0
votes
2answers
25 views

How do I compute this recursive function efficiently? [closed]

Let $f(x,y) = xy + f(x-1,y-1) $ where $f$ equals $0$ if either $x$ or $y$ is $0$. Also $x,y$ belong to $\mathbb{N}$. Describe an efficient (less then $O(n)$) algorithm for computing $f(x,y)$.
2
votes
1answer
26 views

Incorrect Euler Totient Function definition?

According to wikipedia, definition of Euler's totient function (or Euler's totient function) is: Euler's totient function is an arithmetic function that counts the positive integers less than or ...
0
votes
0answers
21 views

Need simple logic or formula for the below problem!

The problem is simple tip calculator here calculating remaining tip from the money got from user. Inputs - x,y,z Where "x,y" are two denominations of currency and "z" is billamount If x = 2, y=5, ...
3
votes
0answers
67 views
+50

Looking for help to clearly define a function that counts the number of twin primes in a range

My goal is to define a function that counts the number of twin primes between $q$ and $q^2$ where $q$ is any prime greater than $7$. I would like to do this using: The Sieve of Eratosthenes The ...
1
vote
2answers
28 views

Difference between domain and range for relations and functions?

What is the difference between the definition of domain and range for a relation, and that for a function?
5
votes
3answers
62 views

“Perimeter” of the sine function

Given a sine function with certain parameters (period, amplitude) I would like a function to calculate its "perimeter", i.e. the length of the curve itself. Everyday application: let's say we need to ...
0
votes
4answers
39 views

What is the exact definition of an Injective Function

Am I right to believe that a function is injective, if some elements of the first set are mapped to some elements of the second set? It is also possible to 4 elements of the first set, are mapped to ...
1
vote
2answers
72 views

Does $ f([0,1])$ must always contain an irrational number?

if $f(x)$ is a one to one function from $[0,1]$ to $[0,1]$ then how to prove that $ f([0,1])$ must contain an irrational number. Also prove that $f(x)$ need not contain rational number. I really ...
2
votes
1answer
35 views

How do I prove this function is differentiable at 0?

Define $f:\mathbb{R}\longrightarrow \mathbb{R}$ by $$f(x) =\begin{cases} x^{4/3}\cos \left(\frac1x\right) & \text{if } x \neq 0, \\\\ 0 & \text{if } x =0. \end{cases}$$ Prove that ...
0
votes
0answers
11 views

Graphs Approaching Asymptotes

I've been wondering this for a while. For graphs that approach asymptotes, are there certain formulas that can determine the distance between the graph and the asymptote as $x$ gets infinitely small ...
2
votes
1answer
58 views

Proof regarding a probability generating function (Poisson)

Let $f(s)$ be the probability generating function ($pgf$) of a non-negative, integer valued random variable. It is also given that $f(1-p+ps)f(p) = f(ps)$. Prove that $f(s) = e^{\lambda(s-1)}$ for ...
0
votes
1answer
19 views

The relation between the entropy of random variables $X$ and $Y=g(X)$

A previous post has shown that for random variables $X$ and $Y=bX$, where $b > 0$, the entropy of $X$ and $Y$ are not equal (Entropy of $Y=bX$). However, wouldn't any bijection $g$ on a random ...
8
votes
0answers
86 views

Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$

Problem: Find all continuous real-valued functions $f$ such that $$f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}.\tag{1}$$ Here $f$ is allowed to be defined only on a subset of $\mathbb{R}$. The only ...
1
vote
0answers
32 views

Is this a $\log,\zeta$ relationship

Is there a relationship between $\zeta(x^{1/x})$ and $x/\log(x)?$ They seem remarkably close in value.
0
votes
0answers
13 views

Maximization of ratio of two polynomials over an interval

I have a function $h(x) = \frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ are polynomials of the form $a(x + b_1)(x + b_2)(x + b_3)...(x + b_n)$ for some constants $a > 0$ and $b_i > 0$. I ...
0
votes
2answers
20 views

Extrema of functions of two variables: necessary and sufficient conditions

I seem to recall my teacher telling us about the necessary and sufficient conditions while finding the maxima/minima of functions. However, I can no longer find those conditions in my booklet and even ...
1
vote
1answer
34 views

Maximum and minimum of a function from $\mathbb{R}^n$ to $\mathbb{R}$

Let $A \in \mathbb{R}^{n \times n}$ be a real $n \times n$-matrix. Consider the function $$q: \mathbb{R}^n \to \mathbb{R}, x \mapsto x^t A x$$ where $x^t$ is the transposed vector $x$. I now want ...
-3
votes
1answer
58 views

Function Exercises [closed]

Prove that no function, $f$, exists such that $f(\sin x)+f(\cos x)=x$