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

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1
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1answer
27 views

How to find a function/operator that satisfies the following conditions

I'm looking for a function that satisfies : 1) Symmetric: $f(x,y) = f(y,x)$ 2) Associative: $f(f(x,y), z) = f(x,f(y,z))$ 3) $f(x,x) = 0$ 4) it would be nice if $f(x,0) = x$, or at least that $g(x) ...
0
votes
1answer
25 views

If $\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ bounded then $\sqrt{\int_a^b(x(t)-y(t))^2\text dt}$ bounded?

Prove or disprove with counter-example: if the set of the functions are bounded at $d(x,y)=\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ then the set also bounded at $d(x,y)=\sqrt{\int_a^b(x(t)-y(t))^2\text ...
1
vote
4answers
113 views

Trouble finding the inverse of $f(x) = x + \frac{1}{x}$ .

Let $ f: \Bbb R - \{0\} \rightarrow \Bbb R \;\text{ given by } f(x) = x + \frac{1}{x} . \text{Find} $ $f(f^{-1}(\Bbb R))$ , $\Bbb R = \text{real numbers}$. For this problem I know one needs to ...
-3
votes
4answers
103 views

Are functions $f(x) = \frac{x^2 + x }{x+1}$ and $g(x) = x$ equal? [closed]

Given $$f(x)= \dfrac{x^2 +x }{x+1} \qquad \qquad \qquad g(x) = x$$ Is it true that $f=g$?
-2
votes
3answers
87 views

Proving why $\Bbb Z_{4} \rightarrow \Bbb Z_{6} \text{ given by } f(\overline x) = [2x+1] $ is not a function. [duplicate]

Question presented: Is following a function from the indicated domain to the indicated co domain? $f:\Bbb Z_{4} \rightarrow \Bbb Z_{6} \text{ given by }$ $ \bbox[white,1px,border:1px solid red]{...
1
vote
1answer
74 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
-1
votes
3answers
45 views

Question involving two functions f and g

f(x)= $\ x^2 $ , $\ x > 2 $ $$ $$ g(x)= $\ x^2 $ , x $ \in $ [0,4] Explain why f and g are different functions
-2
votes
0answers
12 views

Contour plot on MATLAB [closed]

I'm trying to generate a contour plot of this function $$\rho = \frac{1}{2\pi(R^2+(1+\sqrt{z^2+0.4})^2)^{3/2}}$$ using MATLAB. I've defined a mesh grid [R,z] = meshgrid(-10:0.1:10, -10:0.1:10); and ...
1
vote
2answers
40 views

Arc of curve of function - Find the minimum length

This is the arc of the curve: $$y = x^3 + x^2 - \frac{29x}{2} + 1 \\ t > 0 \\ x \in [t,t+1]$$ Find $t$ for which the length of the arc of the curve is minimum. Should I use $ \int {\...
1
vote
2answers
48 views

Prove that this is one-one, but not onto $\Bbb R$.

$\Bbb R$ stands for real numbers. $ f(x) = \begin{cases} 2-x, & \text{if $x \le 1 \qquad \text{is one to one but not onto } \Bbb R $ } \\ \frac{1}{x} , & \text{if $x >1$ } \end{cases}...
1
vote
1answer
84 views

The strange “turning point” of $\arctan(x)/\arctan(\sqrt{x})$

After looking at an interesting graph: $$y=\frac{\arctan(x)}{\arctan(\sqrt x)}$$ There seemed to be a turning point around $(3{,}88;1{,}198)$ (https://www.desmos.com/calculator/58wloddve3) <- A ...
1
vote
2answers
35 views

Notation of the square (or other power) of a function $f(x)$

How do you notate the square (or other power) of a function $f(x)$? Is it $f^2(x)$ (similar to $\sin^2(x)$ for example), $f(x)^2$ or do you have to use $(f(x))^2$? Thanks in advance.
1
vote
0answers
31 views

Continuity of an integral with a removably discontinuous integrand

Say I have a function $f_c:[0,\ell]\to\mathbb{R}$, where $c\in K\subseteq\mathbb{R}^n$, with $K$ compact. For every choice of $c$, $f_c$ is continuous everywhere except at some $s_c\in[0,\ell]$, ...
1
vote
2answers
95 views

The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ [duplicate]

The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ I have no idea how to solve this problem. Can someone point me in the right direction?
5
votes
2answers
306 views

Can the derivative prove my function has only one root?

I have a function: $$f(x)=x-\ln(x^2+1)+2$$ I want to prove my function has exactly one root. If I differentiate: $$f'(x)=1-\frac{2x}{x^2+1}$$ I can see this value is positive for every $x$. Does this ...
6
votes
2answers
43 views

If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$

If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$ Here $[.]$ is for greatest integer function. Attempt: since domain of $f(x)$ is $[-1,2]$ therefore for $f([x]-x^2+4)$...
1
vote
0answers
28 views

Finding functions that give nice solutions to a recurrence relation.

In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\...
0
votes
1answer
50 views

Thought-provoking functional computation problem

I have been assigned a very thought-provoking functional computation problem (to be completed $without$ a calculator) which has left me essentially stumped—that is, I really can't come up with an ...
2
votes
2answers
34 views

Difficult Functions Evaluation Problem

I have a question about finding the value of a certain function that I cannot wrap my head around. The question is: Given a function $f(x)$ satisfying $$f(x) + 2f\left(\frac{1}{1-x}\right) = x,$$ ...
3
votes
7answers
257 views

The map $f:\mathbb{Z}_3 \to \mathbb{Z}_6$ given by $f(x + 3\mathbb{Z}) = x + 6\mathbb{Z}$ is not well-defined

By naming an equivalence class in the domain that is assigned at least two different values prove that the following is not a well defined function. $$f : \Bbb Z_{3} \to \Bbb Z_{6} \;\;\;\text{ ...
2
votes
2answers
55 views

Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
1
vote
3answers
36 views

Trouble proving that this is a function?

By naming an equivalence class in the domain that is assigned at least two different values prove that the following is a well defined function. $$f : \Bbb Z_{3} \to \Bbb Z_{6} \;\;\;\text{ given ...
0
votes
3answers
44 views

Value of $f^2(4)+g^2(4)$ [duplicate]

If $f(x)=g'(x),g(x)=-f'(x)$ for all real $x$ and $f(2)=4=f'(2)$ then value of $f^2(4)+g^2(4)$ is ? Now the above is true when we have a constant function with constant $0$. But then that would not ...
1
vote
3answers
578 views

Prove that a map is Injective

How can I prove that $f: (0, \infty) \times (0,\pi) \to \mathbb{R}^2$ where $f(x,y) = (\sinh(x)\sin(y),\cosh(x)\cos(y))$ is injective?
23
votes
9answers
8k views

Why is an ellipse, hyperbola, and circle not a function?

I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function. But I don't understand ...
2
votes
1answer
77 views

Trouble Finding $f \circ g \; \text{ and } \; g\circ f$ for this function?

$f(x) = \begin{cases} 2x+3, & \text{if x $\lt$ 3} \\[2ex] x^2, & \text{if $x \ge 3$ } \end{cases}$ $,\qquad$ $g(x) = \begin{cases} 7-2x, & \text{if x $\le$ 2 } \\[2ex] x+1, & \...
4
votes
3answers
670 views

What does this dollar sign over arrow in function mapping mean?

In a certain function mapping like this, $x \xleftarrow{\$} \{0,1\}^k$ (Lecture Notes on Cryptography by S. Goldwasser and M. Bellare, page 18) I fail to understand what exactly does this \$ sign ...
10
votes
6answers
15k views

Find formula from values

Is there any "algorithm" or steps to follow to get a formula from a table of values. Example: Using this values: ...
1
vote
2answers
31 views

Trouble Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$

Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$ My problem with this question is that I do not know how one derives the theory in order to get the correct answer. I will ...
0
votes
0answers
10 views

Relationship Between Variables Both Growing/Decreasing, not Proportional

So proportional is when the variables are equal to one another when multiplied by a constant. What is the term for something like weak proportionality that when one variable increases the other will ...
0
votes
2answers
33 views

Solution of composition of function

In a book I saw a question along with solution The question is Let f,g,h be function from R to R , then show that (f+g)oh = (foh).(goh) But when I saw the solution i got confused , they have ...
2
votes
1answer
61 views

Method for solving 2nd order linear PDE of three variables

For the 2nd order linear PDE below, please give method(s) to solve it, working, a solution, and what conditions the solution can exist? $$\sin(t)\frac{\partial^2y}{\partial t^2}+\cos(t)\frac{\partial ...
2
votes
1answer
48 views

Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...
0
votes
0answers
34 views

Is there a general solutions to min(Floor(N/a), Floor((N ± a)/r))?

I am looking for a general solution to $\min\left({\left\lfloor{\frac{N}{a}}\right\rfloor, \left\lfloor{\frac{N \pm a}{r}}\right\rfloor}\right)$ where $N$, $a$, and $r$ are positive integers with the ...
0
votes
2answers
59 views

Discrete mathematics, equivalence relations, functions.

I'd like some insight on how to 'solve' this problem (more towards understanding what the problem is asking) Suppose that $A$ is a nonempty set and $\mathcal{R}$ is an equivalence relation on $A$...
1
vote
0answers
24 views

Have I applied this function to a number properly?

There is a bounded, continuous, even function of a variable t that satisfies the functional equations $$f(t)+f\bigg(t+\frac{1}{2}\bigg)=0$$ and $$2f\bigg(\frac{t}{4}\bigg)+f\bigg(t+\frac{1}{8}\bigg)=1$...
1
vote
0answers
22 views

Construct a master (possibly hypergeometric) formula from a family of formulas indexed by the half-integers and integers

I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a ...
3
votes
1answer
411 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ K(k):=...
3
votes
2answers
43 views

Solution of equation

If $f(x) = x^2 - 2ax + a(a+1)$ , $f:[ a, \infty] \to [a,\infty]$ . If one of the solution of the equation $f(x)=f^{-1}(x)$ is $5049$ , then what may be the other solution ? My WORK: I found the ...
0
votes
2answers
23 views

Mapping of equivalence classes of integers modulo $n$

This is an exercise problem from Essentials of Discrete Mathematics (3rd Edition) by David J. Hunter. The problem is as follows: Consider the function $p : \mathbb{Z} \rightarrow \mathbb{Z}/n$ ...
1
vote
1answer
28 views

Is every strictly increasing function is one to one [closed]

Prove or disprove that every strictly increasing function is one to one
0
votes
0answers
18 views

Generalized exponential and logarithmic functions

The $q$-exponential and $q$-logarithmic functions are defined as in here. Does any one know whether this definition can be extended to $q=\infty$?
2
votes
7answers
108 views

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes but $x^3=y^3$ then $x=y$

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes , but $x^3=y^3$ then $x=y$ holds in the real numbers. . I don't understand a thing If we do like this: $\sqrt{x^2}=\sqrt{y^2}$then $x=y$ but why ...
0
votes
0answers
15 views

Reading a 3d graph to generate a 2d projection.

I know this will sound very dum but I have spent some good time trying to understand before posting this question. Basically, I need some help in understanding how (a) and (b) are being used to ...
4
votes
1answer
881 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
1
vote
2answers
473 views

Characteristic function of union of two sets formula and intuition

From http://topologicalmusings.wordpress.com/2008/03/20/inclusion-exclusion-principle-counting-all-the-objects-outside-the-oval-regions-2/ Is there an easier proof or way to calculate $1[A \cup B]$?...
-2
votes
2answers
63 views

$ g\circ f$ injective $\implies$ $f$ injective or $g$ injective

First I have to prove that: If $g\circ f$ injective $\implies$ $f$ injective or $g$ injective And real functions that improve: If $g\circ f$ injective $\implies$ $f$ injective and $g$ injective I ...
0
votes
1answer
59 views

Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
1
vote
3answers
42 views

$a,b,c$ are Real Number, $(a+1)^{0.5} - a + (b+2)^{0.5} \cdot 2 - b + (c+3)^{0.5} \cdot 3 - c = \frac{19}{2}$, find $a + b + c$ ??

$a,b,c \in \mathbb{R}$, $(a+1)^\frac{1}{2} - a + (b+2)^\frac{1}{2} \cdot 2 - b + (c+3)^\frac{1}{2} \cdot 3 - c = \frac{19}{2}$, find $a + b + c$ ? The answer is: $-\frac{5}{2}$ Please give me some ...
1
vote
1answer
56 views

Solving coupled second order ODEs via Laplace transforms & Function theory.

I have used Laplace transforms to transform a system of 2 coupled second order ODEs into 2 simultaneous equations. 1st ode: $$\frac{3d^2y}{dt^2}+\frac{dy}{dx}=0$$ 2nd ode: $$\frac{5d^2y}{dx^2}-\...