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

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0
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1answer
39 views

Find the range of this function [on hold]

How do you find the range of the following function, please? $$\frac{2x^2 + 20}{x^2 + 5}$$
0
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0answers
8 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
-3
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2answers
30 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
1
vote
3answers
39 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
0
votes
1answer
31 views

Explain why this composite function is not allowed?

Explain why this composite function is not allowed when $f(x) = 2x+1, x \in [-5,5]$ and $g(x) = x^2, x \in \mathbb{R}, x \geq 0$ How would you change the domains so that the function $fg(x)$ can ...
0
votes
0answers
41 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
2
votes
0answers
48 views

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $f(m^2+f(n))=f(m)^2+n.$

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $$f(m^2+f(n))=f(m)^2+n.$$ Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
-2
votes
0answers
43 views

How to prove that composition of functions is a function [on hold]

Using the fact that a function is a relation, which is a subset of the product of $X$ and $Y$. $(a,b)$ belongs to $f$ and $(a,c)$ belongs to $f \implies b=c$
0
votes
0answers
24 views

Piecewise Logistic Function [Satellite Data]

I am working with $16$-day MODIS EVI (satellite) data and I want to fit a Piece-wise Logistic Function through my $23$ EVI data values. The following formula is for the Piece-wise Logistic Function: ...
5
votes
5answers
2k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
1
vote
0answers
53 views

Confused about basic of image

Hello I tried to work a problem from the text called " Introduction to Real Analysis" by Robert G Bartle and Donald Sherbert and I encountered a small difficulty. I am starting to think that my ...
2
votes
4answers
47 views

What does $f^{-1}(B)= \{ x \in X \mid f(x) \in B\}$ mean?

I have encountered the expression $$f^{-1}(B) = \{ x \in X \mid f(x) \in B\}$$ My questions are: 1) What does the $-1$ exponent mean in this context? 2) Is it right to say "if the set $X$ ...
0
votes
1answer
39 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
1
vote
1answer
74 views

Is there a name for the function of a semicircle?

Recently I've learned many different names for different types of functions... but I've been wondering, is there a name for this type of function? $\sqrt{x - x^2}$
4
votes
2answers
32 views

Prove that $f(X\cap f^{-1}(Y))=f(X)\cap Y$

Let $\ f\colon A\to B$ and let $X\subset A$, $Y\subset B$, prove that $$f(X\cap f^{-1}(Y))=f(X)\cap Y$$ The "$\subset$"$-$inclusion is easy: if $y\in f(X\cap f^{-1}(Y))$, exists a $x\in X\cap ...
2
votes
2answers
64 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
3
votes
1answer
243 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
0
votes
1answer
54 views

What is the function $f$ such that $\sum_{k=0}^n f(k)=n^3$?

$$\begin{align*} 1 &\leadsto 1 \\ 1+3 &\leadsto 2^2 \\ 1+3+5 &\leadsto 3^2 \end{align*}$$ In general, if $f(x)=2x+1$, then $f(0)+f(1)+f(2)...f(n)=(n+1)^2$. Now, $$\begin{align*} 1 ...
1
vote
1answer
17 views

Find formula structure for a complex function

I am looking to find the function formula structure of a repeating function like the one in the image linked below.... Something that repeats indefinitely (like a sine wave) on the X-axis. Anybody ...
-1
votes
2answers
61 views

Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$. [on hold]

Let $f(x)$ be a polynomial in $x$ With integer coefficient. If for natural numbers $a,b,c$, $f(a)=b, f(b)=c, f(c)=a$ Prove that $a=b=c$.
0
votes
1answer
34 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
1
vote
2answers
87 views

How to prove even subsets equal to odd subsets? [duplicate]

There is question that I don't know how to prove. we have set $A=\{1,2,3,\ldots,n\},\; O=\{B\mid B⊆A,\text{ odd }B\},\; E=\{B\mid B⊆A,\text{ even }B\}$ it ask to prove that subsets even equal to ...
9
votes
4answers
731 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
1
vote
2answers
43 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
1
vote
3answers
46 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ ...
1
vote
2answers
22 views

Can we say that a function is increasing/decreasing on some range if there's a vertical asymptote in that range?

The graph below shows the function $f(x)=\frac{e^x}{x-1}$ Can we say that the function is decreasing for all $x\le2$ (there's a local minimum at $x=2$) or do we have to take the asymptote at $x=1$ ...
3
votes
8answers
100 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
2
votes
1answer
30 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...
0
votes
1answer
23 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
3
votes
0answers
27 views

Technical name for an almost-monotonic function

I'm wondering if there’s a technical name or short phrase that describes a function that’s monotonic, subject to some uniformly bounded amount of backtracking. $\exists \epsilon \forall x , y : y \gt ...
0
votes
4answers
48 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
0
votes
1answer
63 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
3
votes
3answers
69 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
0
votes
1answer
27 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
3
votes
3answers
36 views

Finding periodic (trigonometric?) function given points

It's been a while since I've taken a math class. I need a couple functions for a program I'm working on. I can tell they involve trigonometry, but I can't figure out how to derive the function ...
1
vote
0answers
23 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
-1
votes
1answer
21 views

Ratio known and Amount needed known

You have a a recipe for the perfect orange juice, it is: 26 parts Water to 1 Part Orange Juice (26:1). However overall you only want 20ml of mixed liquid (orange juice and water combined). What ...
0
votes
0answers
13 views

Function/algorithm to generate a random walk on a graph

I'm looking for a graph function or an algorithm that can generate a random fluctuating random walk that will eventually converge between the value of y = 0 and y = 1, more or less after a number of ...
1
vote
0answers
51 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
0
votes
1answer
35 views

Confused by one-to-one question, I think it's order incorrectly

I have this question and it seems a tad redundant If $A$ and $B$ are infinite sets, is it possible for there to be a 1-1 function from $A$ to $B$ and a 1-1 function from $B$ to $A$ without there ...
0
votes
4answers
66 views

Why is this function a bijection?

Consider the function below $$f:\mathbb{R^+} \to \mathbb{R^+}$$ given by $$f(x) = \sqrt{x}$$. Now it makes sense that the function is injective because $f(x) = f(y) \implies \sqrt{x} = \sqrt{y} ...
1
vote
1answer
112 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
0
votes
1answer
47 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
0
votes
1answer
30 views

$f(x)=y$ while $g(y)=x$; Is it possible to find two not reverse functions that behave such at least for a given set of inputs and outputs?

I want to know if it is possible to program such a code that could determine two distinguish, not inverse, functions, say $f$ and $g$, that is true for the below statements at a given input and output ...
1
vote
3answers
28 views

Given $f(x)= \frac{1}{4}(x+4)^2-2$ Find vertex, $ y$ intercept etc.

Given $f(x)= \frac{1}{4}(x+4)^2-2$ Find: vertex, $y$-intercept, $x$-intercepts (if any), axis of symmetry What I have so far: Vertex: $(-4,-2)$ $y$-intercept: $(0,2)$ $x$-intercept: $2$ Axis of ...
0
votes
1answer
152 views

Differential Calculus Problem - Sphere volume increasing (differentiation of algebraic functions)

The Air is pumped into a spherical ball which expands at a rate of 8cm^3 per second. Find the exact rate of increase of the radius of the ball when the radius is 2 cm. I have tried this question, ...
-11
votes
0answers
24 views

Please write down exact output after the following statements are proceesed [on hold]

Please write down exact output after the following statements are proceesed ...
1
vote
1answer
27 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
0
votes
2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?