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

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2
votes
1answer
570 views

Inserting if else in a mathematical expression is it possible

Hope you guys can help me out. I have a mathematical expression which I constructed its: $$ A_S = \frac{A_{U_{Max}} - A_{U_{Min}}}{U\sqrt{2}} $$ Is there a way in which I could mathematically specify ...
1
vote
4answers
71 views

Does this inequality hold

Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not ...
1
vote
1answer
186 views

How would I express conditional summation.

I was hoping the experts here could help me out. I need to express conditional summation. Let me give you guys an example. Suppose I have a timeline like this: Each value on the time-line in the ...
0
votes
1answer
303 views

a multivariate quadratic function

Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where $$f_1(x,y)= x^2+3xy$$ $$f_2(x,y)= 2xy+y^2$$ (here f is column vector, x, y are variables) Assume that each $f_i$ is a ...
0
votes
1answer
583 views

Prove that the integral of an odd step function on [-1,1] is 0

Let $\varphi\colon[-1,1]\to \mathbb R$ be an odd step function.Prove that: $\int_{-1}^1\! \varphi(t)\, dt = 0$ Thanks!
5
votes
7answers
2k views

Proving a set is uncountable

I'm having a bit of trouble thinking of how to prove this homework problem.Prove that a set $A$ is uncountable if there is an injective function $f:(0, 1)\rightarrow A$. I know $(0, 1)$ is ...
0
votes
3answers
142 views

Trying to figure out a simple linear function

Maximum input to function is $40$, minimum is $1$. Function works like this: $f(40)=0.2$, $f(1)=1$ we also want to know what $f(x)$ would produce. The answer would be "linear" to these two ...
0
votes
1answer
538 views

Linear homogeneous recurrence relation with constant coefficients: How does one determine the solution set?

According to my textbook and this Wikipedia article, a recurrence relation of the form $$ b_0 a_n + b_1 a_{n-1} + \cdots + b_k a_{n-k} = 0 $$ (EDIT: where $ b_0 \neq 0 $) has the following set of ...
2
votes
2answers
124 views

How to measure how erratic is a function between a and b

I need to compare the outputs of some functions and to rate their "erratness". Given a function, the less erratic function between a and b would be a straight line and the more erratic function ...
1
vote
5answers
362 views

Simple formula to find number with all consecutive digits from 1 to $x$?

Given number of digits required, $x$, find an $x$-digit number such that $f(1) = 1$, $f(2) = 12$, $f(3) = 123$, $f(4) = 1,234$, and so forth. I'm banging my head against the wall trying to ...
0
votes
0answers
100 views

Given an Associative Operation on an Infinite Set, How Many Similar Operations are There?

With respect to an operation $O_n$ on an infinite set, define a mimic operation $O'_n$, as an operation which differs from $O_n$ only by a finite number of points, and a $k$-mimic operation $O^{k}_n$ ...
0
votes
2answers
169 views

Linearity Of A Function

I understand that the linearity of a function is determined by the degree of the polynomial but I was unsure whether the modulus operator changes this? Is $f(x)$ = N mod x a linear function if $N$ ...
-3
votes
1answer
155 views

Function collection with specific properties

I'm looking for at least one answer to each number. If I know some function that holds I will put the answer, but, if there others, I would like to know too. If is there's a similar function with the ...
1
vote
1answer
140 views

Currying for dependent functions

Currying and uncurrying is defined between functions in $Z^{X \times Y}$ (the first set) and $\left( Z^Y \right)^X$ (the second set). But what if $Y$ is not a constant but is dependent on $X$? The ...
1
vote
1answer
418 views

mixed random variable cdf

Let $$F(x) =\cases{ 0, & $ x\lt0$ \cr x^2+0.2, & $0\le x\lt 0.5 $\cr x,& $0.5\le x\lt 1$ \cr 1,& $x\ge 1$. }$$ How do I rewrite $F(x)$ like ...
4
votes
0answers
142 views

Finding a series of functions orthogonal to all $x^{2n}$ but one

We need to find a series of functions $f_m$ verifying the property $$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$ We already have found $$f_0(x) = ...
0
votes
1answer
167 views

Is there any *useful* $f$ who satisfies this properties?

Is there a useful (see below to understand bit more what is useful) function with the properties: $$f(x)=0 \quad x< k$$ and $$f(x)=1 \quad x\geq k$$ to $$k \in \mathbb{R}$$ But it couldn't be a ...
2
votes
1answer
74 views

What is a “normal value”, and why do I need it to slice a pie?

I am working on a Pie Chart, and am using a tutorial that states… An easy way to get the relative value is to normalize the array and use the normal value ...
1
vote
1answer
355 views

How to denote a function mapping of two parameters to an output?

I am wondering how to denote something like this: I have a function mapping two sets to a set of vectors and am currently denoting it as so: $p: [F, \Sigma] \to S $ I am wondering if this is ...
2
votes
1answer
343 views

Function with the same derivative are equal up to a constant

If functions $f$ and $g$ are continuous on $[a,b]$ differentiable on $(a,b)$, and $f'(x) = g'(x)$ on $(a,b)$, then there exists a real number $K$ such that $f(x) = g(x) + K$ for all $x\in [a,b]$.
1
vote
3answers
383 views

Can there be a scalar function with a vector variable?

Can I define a scalar function which has a vector-valued argument? For example, let $U$ be a potential function in 3-D, and its variable is ...
1
vote
1answer
345 views

Are there any implicit, continuous, non-differentiable functions?

Like the title suggests. Is it possible to have an implicit function that is continuous but not differentiable? Something which resembles a fractal, or is perhaps constant (not asymptotic) after a ...
0
votes
1answer
247 views

Is the derivative of a Lipschitz function Riemann integrable?

Suppose $f:\mathbb R^{n}\to \mathbb R$ is a Lipschitz function. Is $\sqrt{1+|\nabla f|^2}$ Riemann (not Lebesgue) integrable on a bounded open set, say a ball? In $\mathbb R^1$, a function is ...
4
votes
1answer
173 views

How can I find power series of $f(x)$?

$$f(x)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x^2}{1+\dfrac{x^3}{1+\dfrac{x^4}{\ddots}}}}}$$ How can a power series be found given the continued fraction $f(x)$? I'm trying to find $f(x) ...
0
votes
2answers
57 views

Class of functions to combine several parameters into some “satisfaction index”

I'm sure the title itself is puzzling. But my problem is quite simple: I'm looking for a class of functions, that can combine several variables(all of them are within known ranges), given their values ...
0
votes
1answer
276 views

Asymptotic notation - some equations.

i have a problem with proof one of this facts: $2^{2^n}$ = $\Theta (n^n)$ or $2^{2^n}$ = $O (n^n)$ or $2^{2^n}$ = $\Omega (n^n)$ and to proof one of this: $(n^n)$ = $\Theta (2^{2^n})$ $(n^n)$ = ...
2
votes
3answers
341 views

Square Root Of A Square Root Of A Square Root

Is there some way to determine how many times one must root a number and its subsequent roots until it is equal to the square root of two or of the root of a number less than two? sqrt(16)=4 ...
1
vote
0answers
164 views

Solving functions of the form $f(x)=g(f(h(x)))$

Suppose we have say $$f(2x)=\left(f(x)\right)^4,$$ with $f(0)=1$ and $f$ not being constant. How does one find out what $f$ is without guessing? More generally, is there a systematic way of finding ...
0
votes
1answer
52 views

Finding the equation of a line

I'm using this formula to work out the equation of a line joining points $(1,5)$ and $(-9,2)$: $$ \frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1}$$ Like this: $$ \frac{y-5}{2-5} = \frac{x-1}{-9-1}$$ ...
-1
votes
2answers
164 views

I can't find a absolute value function that have [-1,1] range

I want a function $f:\mathbb{R}\to[-1,1]$ with absolute value like $f(x)=|a-x|\ldots$ that have $[-1,1]$ range. Can anybody help me?
3
votes
2answers
778 views

If $f(x + 1) + f(x − 1) = f(x), \forall x \in \mathbb{R}$,then how to find $k$ such that $f(x + k) = f(x)$?

Let $f(x)$ be a function such that $f(x + 1) + f(x − 1) = f(x), \forall x \in \mathbb{R}$. Then for what value of $k$ is the relation $f(x + k) = f(x)$ necessarily true for every real $x$? The ...
-2
votes
2answers
187 views

Simple ceiling function problem [closed]

Prove that $\lceil4n/3\rceil\le 4\lceil n/3\rceil$ for all integers $n$. Try to generalize this result to something where something other than 4 and 3 are used.
0
votes
2answers
131 views

How to plot this function

I want to plot functions (using some software, any recommendations?) that looks like this below. Could someone suggest equations of functions that would look like the graph below?
0
votes
1answer
2k views

Finding minimum/maximum of a multi-variable function under some constraints using matlab.

I have a function $f(x,y,z)$ I want to find the minimum/maximum of the function with some constraints like $$0 < x < C_1$$ $$0 < y+z <C_2$$ where $C_1$ and $C_2$ are some integer ...
0
votes
2answers
91 views

Simple question about the ceiling function

Is $n\lceil x \rceil = \lceil nx \rceil$, for all integers $n$ and real numbers $x$? If not, are there some similar (or less general) rules available? If yes, can this somehow be generalized?
0
votes
1answer
96 views

How to convert the constants in a regression equation to constants in a linear equation

Hello dear mathematicians, I'm not entirely sure what to tag this question with since I'm new here but I hope some more experienced user can guide me. Here is my problem: I'm using an internal ...
1
vote
1answer
133 views

Counting number of functions

I am reading the paper here and have a small doubt in Lemma 1. The proof (on page 1) begins with: Lemma 1. Let $M$ and $N$ be cardinal numbers. Let $S$ be $N$-regular in $X$, a set of ...
3
votes
1answer
313 views

How is this function additive?

Linear functions are said to be additive: $f(x + y) = f(x) + f(y)$ But if I have this simple function $f(x)= 7x+3$, I get, for example(at $x=5$ and $8$): $f(5)=38$ and $f(8)= 59$. The sum is $97$. ...
5
votes
6answers
4k views

Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?

It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.
6
votes
3answers
1k views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
1
vote
2answers
102 views

Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?

This question is a by-product of this one. I'm asking it because of this comment by Tara B. I'll repeat the definitions. The full transformation semigroup $\mathscr T_X$ on a set $X$ is the semigroup ...
0
votes
1answer
102 views

few “easy” questions about function

Examine the function $y = x^2 - 4x + 3$ and determine: if the curve has a maximum or minimum point? the function's zeroes the function's line of symmetry, coordinates of the turning point
2
votes
1answer
137 views

Algorithms for deciding whether a function over a finite ring is polynomial or not?

Let $R$ be a finite ring, and $f$ be a function from $R$ to $R.$ Suppose I want to know whether $f$ can be represented as a polynomial or not? Are there any good algorithms for finding this out?
0
votes
2answers
298 views

Designing a game score function with non-linear graph

I am trying to create a scoring scale for a game. I want the scale to be non-linear to reflect the realness of the game. The score is determined by time, the less time you take, the greater score you ...
1
vote
1answer
102 views

Continuity of two functions

Well, I am little confused about these problems; I need some help: Show that the equation $$4x-3\cos(x)=1-2t\cos(t)$$ defines a unique continuous function on any $[a,b].$ same problem for the ...
1
vote
3answers
637 views

Help me understand a 3d graph

I've just seen this graph and while it's isn't the first 3d graph I've seen, as a math "noob" I never thought how these graphs are plotted. I can draw 2d graphs on paper by marking the input and ...
3
votes
3answers
231 views

What is the fractional part function of $e^x$?

Given a real positive number $x\in\mathbb{R^+}$. What is the function of the fractional part of $e^x$?
0
votes
3answers
305 views

How would you mathematically represent the master key and 'slave lock' model?

Consider a dormitory building with $N$ rooms $R_1$ to $R_N$. The $i^\text{th}\text{ resident }(i \in [N])$ has a key $K_i$ to his own room $R_{i}$. Let $L_{i}$ be the lock on room $R_{i}$. Naturally, ...
1
vote
1answer
228 views

Why is the differentiability of a piecewise defined function studied using the definition of derivative?

Why is the differentiability of a piecewise defined function often studied using the definition of derivative? For example, let: $$\begin{align*} f(x) &= x\cdot |x-1|\\ &= \left\{ ...
1
vote
1answer
165 views

What does $~ u(\cdot, t)$ mean when referring to a function?

I sometimes stumble over professors defining a function $u$ using regular (but quite sloppy) notation like $u(x,t) = A\sin(x)e^{-kt}$. Later in their notes, they state something like $u(\cdot, t)$ = ...