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

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Determine null, extreme and inflection points of function $f(x)=\frac{x+e^x}{x-e^x}$

This function has a null point, but I can't compute it from equation $f(x)=0$ which gives $$\frac{x+e^x}{x-e^x}=0$$ $$x+e^x=0$$ How to compute this equation? Extreme points can be computed from ...
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1answer
35 views

There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
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1answer
27 views

Drawing a graph of a function.

$h_{1}=pq-\frac{1}{2}kq^{2},\ h_{2}=pq-kq^{2}, \frac{dh_{2}}{dh_{1}}=\frac{p-2kq}{p-kq}$, $k,p$ are constant. My question are how can I draw a graph of function $h_{2}$ measuring $h_{1}$ on the ...
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4answers
149 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
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1answer
37 views

Equality between functions?

Given are real functions as follows: $f_{1}(x)=x, \; f_{2}(x)=\frac{x^2}{x}, \; f_{3}(x)=\sqrt{x^2}, \; f_{4}(x)=\left (\sqrt{x} \right )^2$ Are there any equal among them? I checked the domains ...
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0answers
7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
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1answer
27 views

Why not an Absolute maximum in an open interval?

The function $x^3+x^2\: \text{has a maximun value at}\: x=-\frac{2}{3} \text{in (-1, 0) }.$ My question is why call it a Local Maximun and not an Absolute Maximum when it is the highest value in that ...
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1answer
34 views

A question on vector subspace [duplicate]

Let $V$ be the vector space of all functions $f \colon \mathbb{R} \to \mathbb{R}$ over $\mathbb{R}$, is the set of functions which are continuous a subspace? I think if you add functions which are ...
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1answer
21 views

What is the difference between functions and operations?

Wikipedia says that an operation $\omega$ is a function of the form $\omega: V \to Y$, where $V \subset X_1 \times\cdots\times X_k$. But as far as I know, every function's domain is a set, so ...
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1answer
20 views

Onto (surjective) functions of 2 variables [closed]

I have a couple of functions I'm curious about: $f(m,n)=m^2 -n^2$ and $f(m,n)=|m|-|n| $, for $m,n\in \mathbb{Z} $. The codomain also consists of all integers. My understanding is that for this ...
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1answer
15 views

Is it possible to express the indicator function for a real interval in terms of other function/s?

Suppose that I've an indicator function defined for an interval in $\mathbb R$, i.e., suppose that $f(x)=1$ if $x\in(a,b)$ and $f(x)=0$ otherwise. Then can I express $f$ without using indicator ...
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2answers
77 views

Is 'clamp' a formally recognized mathematical function?

I was surprised to find the clamp function absent from Mathworld and Wikipedia. Is this mainly a concept particular to computer programming? Is it known by another ...
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2answers
30 views

Is the composing of functions always commutative?

I have a question for my math study. It seems quite simple, but I just can't find a counterexample for the following: The composition of two functions is always commutative Could you help me with ...
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1answer
50 views

Why is the following true? functions

$$x , x_0 \in [a,b]$$ $x_0$-fixed $f \in D(a,b)$- differentiable on [a,b] $$\triangle (x)=f(x)-f(x_0)-f'(x_0)(x-x_0)$$ $$\triangle '(x)=f'(x)-f'(x_0)$$
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0answers
78 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
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4answers
64 views

If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one?

The title says it all - but to reiterate: If neither of $f: A \to B$ or $g:B \to C$ is one-to-one can $ g \circ f$ be one-to-one? I think not. Anyone have a good proof for this? This is simply ...
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3answers
24 views

Function and Domain with Deleted Neighbour - Beginner Question

I have a simple question about functions and domains. Consider the following function: $$f(x) = \frac{ x^2-9}{x-3}$$ I often see in the textbooks mentioning that the domain of this function can be ...
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1answer
44 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
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1answer
61 views

Inverse function $g^{-1}$

The function $g$ is defined by $$g(x)= 3-2x-4x^2, x\in \mathbb{R},x\leq -\frac{1}{4} $$ Find the inverse function $g^{-1}$. Calculate the value of $x$ for which $g(x)=g^{-1}x$. My attempt, ...
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0answers
15 views

Extension of a quasiconvex function

A function $f$ defined on a subset $S$ of the $n$ dimensional Euclidean space is said to be quasiconvex if $f(ax + (1-a)y) \le\max \{f(x) , f(y)\}$ for all $x, y \in S, a \in [0,1]$. Now, suppose ...
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1answer
10 views

Convex combination of quasiconvex functions.

Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?
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2answers
22 views

Product of two continuous, non-negative and monotone non-decreasing function is itself..

My question is simple: is the product of two continuous, non-negative and monotone non-decreasing function itself a continuous, non-negative and monotone non-decreasing function? I believe the ...
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1answer
17 views

Showing the following sequence is monotone decreasing

Let $T$ be fixed and define the functions $$a_k(t) = \frac{e^{\mu_k (T-t)} - e^{-\mu_k(T-t)}}{e^{\mu_k T}- e^{-\mu_k T}}$$ for $t \in [0,T]$. Given that $\mu_k$ is a monotonically increasing ...
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1answer
37 views

Function that Represents Divergent Power Series?

Suppose we have the following power series $$\sum_{k=0}^\infty\left(x^2+1\right)^{2k}$$ If we wished to find the function that represents this series, it seems reasonable to suppose that the ...
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2answers
20 views

Find all critical numbers of $f(\theta) =\cos(2\theta), 0 < \theta < \frac{\pi}{4}$ then use first derivative test to find rel max or min

The theta and cosine are really throwing me off. I had one of my friends help me out, but the work really does not make sense to me. They got $\sin(0) = 0$ is the only critical number and there is no ...
3
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1answer
33 views

Degree Of Polynomial Factored Function $g(x) = 0.5x (x+4)^2(2x-3)$

I'm confused about the process of how to find the degree of a polynomial factored function. I'm not sure that in this specific question if there is only two zeros/factors or if the 0.5x also plays ...
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0answers
45 views

Given x,y,w,h can you generate a rainbow box/cuboid with rounded edges?

Given $x$, $y$, $w$, $h$ where $0 \leq x < w$ and $0 \leq y < h$ and $(x, y)=(0, 0)$ is bottom-left and $(x, y)=(w-1, h-1)$ is top-right and they're all integers, can you make a formula that ...
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1answer
13 views

Implement a y-ceiling on a slope function without domains

I'm working on a computer software application, dealing with very large quantities of numeric data, and I'd prefer to remove the domain conditions from this graph to help boost performance. Right now ...
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5answers
264 views

Theoretical function question

Suppose we have the function $f(x)= x^2 $. This function associates real numbers with real numbers ( $f:\mathbb{R}\rightarrow \mathbb{R}$). Now, what i get confused sometimes is what exactly the ...
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1answer
16 views

How to prove that a function is convex.

Let $h:[0,1] \to (0,\infty)$ be a continuously differentiable function such that the following inequality is true: $$\frac{h'(t)}{h(t)} > -\frac{1}{2} \ \ \ \ \text{for all $t\in (0,1)$} .$$ Let ...
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2answers
23 views

Understanding functions of matrices

Given $$f(X) = rank(X) $$ with X being a matrix. Is it possible to visualize such a function? What is the space that it lives in (assuming all entries live in $\mathbb{R}$)? Is there literature on ...
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1answer
25 views

Suppose $f\leq g,h$. If $\{ f<g\} \cap \{ f < h \} = \emptyset$, then $\{f<g \} \cap \overline{\{f < h\}} = \emptyset$.

Suppose $X$ is a Hausdorff topological space. Assume that $f,g:X \rightarrow \mathbb{R}$ are continuous functions such that $f \leq g$. Denote $\{f< g \}=\{ x \in X: f(x)<g(x) \}$. Lemma: ...
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1answer
27 views

Show $\phi(x) = (x - x_1)(x - x_2) \cdots (x - x_m)$ is odd for $m$ odd.

Given the function $$ \phi(x) = (x - x_1) (x - x_2) \cdots (x - x_m) $$ where $m$ is odd, and the points $x_1, x_2, \cdots, x_m$ are symmetric wrt the midpoint of its domain, show that the function ...
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1answer
31 views

Proving that a function is increasing

I have this problem Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, such that its Taylor series convergers to function $f$ everywhere. For every derivative of the function $f$ we have that ...
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0answers
49 views

Fixed Points of Function from Rationals to Reals

Consider a function $f$ from the positive rationals to the reals such that $f(x)f(y)\ge f(xy)$ and $f(x+y)\ge f(x)+f(y)$. Further assume this function has a fixed point greater than $1$. Prove this ...
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0answers
21 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
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2answers
35 views

If $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=Id_X$

How to prove that if $f\colon X\to Y$ is injective there is a $g\colon Y\to X$ such that $g\circ f=\operatorname{id}_X$. I know that it is an if and only if, but I have already proved the reciprocal. ...
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2answers
57 views

A difficult problem on functions

I've been trying to solve the following problem but can't wrap my head around it. Let $x$, $f(x)$, $a$, $b$ be positive integers. Furthermore, if $a > b$, then $f(a) > f(b)$. Now, if $f(f(x)) = ...
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2answers
65 views

Derivative of function $f(x) = \sqrt{2x}+ \sqrt{2/x}$

The derivative of function $$f(x) = \sqrt{2x}+ \sqrt{2/x}$$ Here's what I did, $$f(x) = \sqrt{2x}+ \sqrt{2/x} \\ = (2x)^{1\over2} + ({2\over x})^{1 \over 2}\\\\$$ $$f'(x)={1\over 2}(2x)^{-{1\over ...
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1answer
22 views

A continuity result

Suppose (i) $f:R^n_+\to R$ and (ii) $f(x)=f(\alpha x)$, for all $\alpha>0$ and (iii) For any $x,y\in R^n_+$, if $x_n\to x^*$, $y_n\to x^{*}$, we have $\lim f(x_n)=\lim f(y_n)$. Could I claim ...
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0answers
26 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
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1answer
33 views

Function and Derivatives

If $$F(x)= (x-1)^{20} - (x-2)^{30} \cdot(x-3)^{40}$$ The number of real roots of $F''(x)=0$ are? $F''(x)$ - Second Derivative of $F(x)$. I have worked it out by simply differentiating it and then ...
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1answer
30 views

If a function $f$ is invertible can I say that $f^{-1}$ is also one to one and onto?

If we have a function $f$ that is both one-to-one and onto (so it's invertible). Its inverse function $f^{-1}$ is also one-to-one and onto? If this is not true can someone please explain it to me or ...
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2answers
56 views

How to show that $f_{n+1} \leq f_n$?

Let $$f_n(t) = \frac{e^{nt}-e^{-nt}}{e^{nT}-e^{-nT}}$$ be defined on $t \in [0,T]$. Can someone help me, I need to prove that $f_{n+1}(t) \leq f_n(t)$ for every $t$. I tried taking ratios and/or ...
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2answers
30 views

Question about functions and topology

This is a very general question, but one that I have been struggling with. If we say that a function from a topological space X to a topological space Y is ONTO, then does that mean that for each open ...
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0answers
18 views

How to rotate a 2nd derivative gaussian function?

I have a 2nd gaussian derivative in y and a normal gaussian in x, which results in the function: $$ f\left ( x,y \right ) =\frac{- \exp ^{-\left (\frac{x^{2} +y^{2}}{2\cdot \sigma^{2} } \right ...
0
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1answer
25 views

“Positively homogeneous of degree zero”

I am trying to understand a statement in an economics paper (and this paper is unfortunately quite sloppily written). Let $A$ be a finite set. Let $S$ be the set of real-valued functions on $A$, i.e. ...
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1answer
11 views

Gaussian blur filter weight calculation formula is not clear

Here you can see Gaussian blur filter weight calculation formula: ...
2
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1answer
46 views

Question on construction of entire functions

Suppose that $x_i$ and $y_i$ are sequences in $\mathbb{C}$. Can you construct a non constant entire function such that $f(x_i)=y_i$? What happens if $x_i$ have an accumulation point? or what happens ...
0
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0answers
18 views

function notation: parameters of transformed functions

given the function $f(x)=x^6$, what are the parameters of the transformed function $g(x)=\frac{1}{2}\left(3-\frac{1}{2}x\right)^6-2$, what is the effect of each parameter on the graph of the original ...