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

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0
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9 views

What is a parametrized function?

I am readimg this article: Stochastic Gradient Descent Tricks and I would like some precisions: Each example $z$ is a pair $(x, y)$ composed of an arbitrary input $x$ and a scalar output $y$. We ...
0
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2answers
79 views
+50

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
0
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0answers
13 views

Can this multivariable function exist?

(3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles $x^2$ + $y^2$ = 4 and $x^2$ + $y^2$ = 10? If so, what is an example? If not, why not? I initially ...
79
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0answers
3k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
0
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1answer
684 views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
6
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1answer
1k views

Meaning of different Orders of Derivative

I have been trying to analyse the meaning of higher order derivatives and their geometrical significance. Given a function $f(x)$ what are the unique geometric interpretation of its higher orders? ...
0
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0answers
22 views

Can't Understand Graphing this Function

I'm graphing a function, $1/2(4-2x)^{1/2} + 1$, and I'm establishing the following: My parent function: $x^{1/2}$ $(x, f(x))\longrightarrow(-(1/2)x - 4, (1/2)y + 1)$ I don't understand why when I ...
2
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1answer
38 views

Equivalent of $\int_0^{\pi/2}\cos^n(\sin(x))dx$

Let $\displaystyle u_n=\int_0^{\pi/2}\cos^n(\sin(x))dx$. How can I find an equivalent of $u_n$ when $n\to\infty$ ?
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1answer
13 views

How to verify if this is a autocovariance function?

Is this$$γ(h) = 1(h = 0) − 0.5 · 1(|h| = 2) − 0.25 · 1(|h| = 3)$$an autocovariance function? How to check this? Is there a method one can use to check if a given function is an autocovariance ...
1
vote
1answer
47 views

Equivalent of the sum $\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$

Let's consider $\displaystyle f(x)=\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$. Where $f$ is defined, can we find a closed form for $f(x)$ ? What would be an equivalent of $f$ near $1^-$ ?
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1answer
25 views

Sense of the graph of a function

What makes it necessary to define the graph of a function $f:A\rightarrow B$ as $$\{(x,f(x))\mid x\in A\}$$ which makes it a subset of $A\times B$, when this is equal to the function itself, which is ...
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2answers
1k views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
1
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0answers
26 views

Topology over $C^0(\mathbb{R})$

Let $C^0(\mathbb{R})$ be the set of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$, For any continuous function $h > 0$ consider $B_f(h) = \{ g \in C^0(\mathbb{R}) : |f(x) - g(x) ...
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0answers
52 views
+50

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
2
votes
1answer
17 views

how to calculate this logarithmic function?

Im having trouble in graphing this log function: $y=\log _{1/4}\left|x^2-5x+6\right|$ I found the intervals: $(-\infty, 2)$, $(2,3)$, $(3,\infty)$ Should I just give $x$ values and find $y$ to graph ...
0
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0answers
16 views

Finding maximum of convex function (appliance of derivatives)

The task goes as following: Divide the length of $14$ into parts $a$ and $b$, in a way that the sum of surfaces of two squares (which sizes are $a$ and $b$), is minimal. $14=a+b => b=14-a$ ...
1
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3answers
5k views

Given $f(x)$ its inverse function, domain and range

$f(x) = \frac{{2x + 3}}{{x - 1}},\left[ {x \in {R},x > 1} \right]$ I've got the inverse function to be: ${f^{ - 1}}(x) = \frac{{x + 3}}{{x - 2}}$ How would I go about working out the range and ...
0
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0answers
14 views

Prove that enumerable set of complex exponentials is linear independent

Define $f_j(p) = e^{i u_j \cdot p}$ for $j=1,2,3,...$, $u_j, p \in \mathbb{C}^N$, $i = \sqrt{-1}$ and $\cdot$ is the scalar product. I need help to prove that the set $\{f_j : j=1,2,...\}$ is linearly ...
0
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1answer
42 views

what is going on here?

Suppose we have a function $f(x), D:( -\infty,0)\cup (0,\infty)$ and for which $$f'(x) = \frac{x^3-1}{x^3} $$ Apparently there is only one point of extremum here, $x=1$, however upon reviewing the ...
3
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2answers
854 views
+100

show that: $f$ is injective $\iff$ there exists a $g: Y\rightarrow X$ such that $g \circ f = idX$

** proof under construction - will post when done and more or less confident it's true. ** also please easy with the downgrades.. i don't understand why i'm getting them. what is meant by show ...
2
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2answers
34 views

Uniform convergence to 0

Let $(f_n)_\mathbb{N}$ be a sequence of continuous functions $[0,1]\to\mathbb{R}$ converging to $0$. The functions are such that for all $x$, $(f_n(x))_\mathbb{N}$ is decreasing. How can one show ...
-1
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0answers
22 views

Inverse function of given statement

we have: $h(x)=(1/2)f(3x)$ what is Inverse function of h(x)? I try this: $3x=t$ $x=t/3$ $h(t/3)=(1/2)f(t)$
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1answer
12 views

Derivative of a distribution

$\DeclareMathOperator{\vp}{v.p.}$ We define $\vp \frac 1x \in \mathcal D'(\mathbb R)$ (the principal value of $\frac 1x$) as $$\left\langle \vp \frac 1x, \varphi \right \rangle = \lim_{\varepsilon ...
14
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13answers
2k views

How to explain the perpendicularity of two lines to a High School student?

Today I was teaching my friend from High School about linear functions. One of the exercises we had to do was finding equations of perpendicular and parallel lines. Explaining parallel equations was ...
0
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2answers
28 views

Why is $f(x,y) = 1/(x^2 + y^2 + 1)$ undefined for the y axis?

I was told that $f(x,y) = 1/(x^2 + y^2 + 1)$ is undefined for the y axis. I.e $x=0$ At first this made sense, but wouldn't the function simply be $f(0,y) = 1/( y^2 + 1)$ which the denominator is not ...
4
votes
1answer
274 views

Function that is discontinuous only for integer fractions

I have this question: Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
0
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2answers
50 views

What function to use to show one to one correspondence?

This problem is from Discrete Mathematics and its Applications Here's an example problem that the author gave I'am working on problem 2e. I first recognized the set as countably infinite. If you ...
0
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2answers
42 views

Finding all continuous functions so that $f^n(x)=x$ for some $n$.

I came up with this problem in class but I can't seem to solve it. I need to find all the functions $f$ with domain and codomain $\mathbb R$ such that there is an $n$ such that $f^n(x)=x$ for all $x$, ...
0
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0answers
60 views

some discrete mathematics problems from sets and functions , [on hold]

I've Been studying for my final exam and i encountered to some problems i could't find the answers , i hope you can help me with any of them . here are the questions : 1 : think that u is the ...
2
votes
1answer
35 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
7
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0answers
69 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
1
vote
1answer
17 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...
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2answers
118 views
+50

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
6
votes
1answer
28 views

Let $A$ be any subset of $\mathbb R^{+}$ , then there exist a metric space $(X,d)$ such that $d:X \times X \to A \cup \{0\}$ is a surjection?

Let $A$ be any subset of the set of positive real numbers $\mathbb{R}_+$ ; then does there exist a metric space $(X,d)$ such that $d\colon X \times X \to A\cup\{0\}$ is a surjection ?
2
votes
5answers
690 views

Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
1
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2answers
42 views

Prove that $f$ is NOT surjective

Let $f: Z \times Z \to Z \times Z$ defined like this: $f(x,y) = (x+y, x-y)$ Prove that $f$ is injective, and not surjective. For injectivity I did that: Let $(a,b) \in Z\times Z$ and $(c,d) \in ...
0
votes
2answers
35 views

If $|B\times A| = 15$ ,evaluate: $|A\cap B|$

If $|B\times A| = 15$ and $|A\times B \backslash B \times B| = 12$. Evaluate: $|A\cap B|$ I tried for myself and got to the conclusion that $|A\times B \cap B \times B| = 3 $ I couldn't get by ...
4
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0answers
124 views

How do you call functions that fulfill $f(x)=\pm f(\pm 1/x)$?

A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd. How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? ...
3
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2answers
58 views

Can we give a bound on any associative function?

We say that $f:[1,\infty)^2\to[1,\infty)$ is associative if $$f(f(a,b),c)=f(a,f(b,c))$$ And symmetric if $$f(a,b)=f(b,a)$$ e.g. the arithmetic operations '+' and '$\cdot$' are associative and ...
1
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1answer
22 views

What are spatial functions?

I was reading Einstein's paper 'Concerning an Heuristic Point of View Toward the Emission and Transformation of Light' and read came across this segment: "While we consider the state of a body to be ...
0
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1answer
32 views

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
-2
votes
5answers
717 views

Solving the equation $\sin t = -\sqrt{2}/2$

Solving the equation $$ \sin t = -\frac{\sqrt{2} }{2} .$$ I know the solution is $1.25$ and $1.75$, but I do not know how to get there. An explanation would be GREATLY appreciated, thanks!
1
vote
2answers
25 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
0
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1answer
19 views

Need help understanding onto function

Let function $g$ from $V = \{1,2,3,4\}$ into V be defined by: $g(n)=3$. I'm having trouble understanding why $g$ is not onto. I understand why it is not one-to-one but, since all the $y$ in $Y$, are ...
0
votes
1answer
25 views

Proving the existence of a Bijection between Cartesian Products of Sets by Induction

Prove by induction that for any sets $A_1, \ldots , A_n$, there is a bijection from $(((A_1 \times A_2) \times A_3) \times \ldots \times A_n)$ to $A_1 \times (A_2 \times ( \ldots (A_{n-1} \times A_n) ...
0
votes
0answers
4 views

Order of Dilated horizontally and translated horizontally

I have a parent function $f(x) = x^2$, and $g(x) = (6[x-2]))^2$ is a transformation from $f(x)$. The question is: $g(x)$ is from $f(x)$ by Dilated horizontally by a factor of 1/6, then translated ...
0
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1answer
65 views

The set of all fixed points of a continuous function $f:[0,1] \to [0,1]$ , satisfying $f \circ f=f$ , is a non-empty interval?

Let $f:[0,1] \to [0,1]$ be a continuous function such that $f \circ f=f$ on $[0,1]$ , then is it true that the set $\{x \in [0,1] : f(x)=x \}$ is a non-empty interval? I can show that it is ...
0
votes
0answers
19 views

Get a function (equation) from data points?

Is there a way to get a function (equation) from data points? For example, we have this famous Google's 'Batman' function: ...
0
votes
0answers
25 views

How do I specify a function without a defined argument?

A function $f$ with the argument $x$ is commonly written $f_x : A\to B, x\mapsto f(x)$, or $f_x : \mathbb{R} \to \mathbb{R}, x\mapsto x^2$, but say I don't want to specify the argument, how would I ...
1
vote
1answer
46 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...