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

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0
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1answer
18 views

Solving function notations.

A gas station attracts customers by offering coupons worth a 0.05 discount for every $1.00 spent on gasoline. a. Complete the table. The equation that represents the relation, using the ...
0
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1answer
44 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
-4
votes
0answers
34 views

What function is this? [on hold]

guys . What function do these pairs represent ? (x , y) : (-1 , -0.5) , (0 , 1.5) , (1 , -2) , ( 2 , +4)
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vote
2answers
36 views

Solving functional equation gives incorrect function

Let $f:\mathbb{R} \to \mathbb{R}$ be a function which satisfies $e^xf(y)+e^yf(x)=2e^{x+y}-e^{x-y}$ for all real x and y. If I place $x=y$, I get $f(x)=e^x-\frac{1}{2}e^{-x}$ which does not satisfy the ...
1
vote
1answer
23 views

Partial vs. complete definition of a function

Suppose I define a function as $f(x,y)=2(x+y)$. Compare that definition to $f:\mathbb{R}^{2}\rightarrow\mathbb{R}, f(x,y)=2(x+y)$, which also gives the (co-)domain. Is there any standard way to refer ...
1
vote
2answers
53 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
2
votes
2answers
564 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
0
votes
3answers
160 views

Create a function where $f(1000)=1.99$ and where $f(400000)=0.49$

As per object I need to create a continuous and decreasing function with one variable where result is as above, but not only. Function should also be easy to modify to get $1<f(1000)<10$ and ...
3
votes
3answers
59 views

Consider the function f(x)=sin(x) in the interval x=[π/4,7π/4]. The number and location(s) of the local minima of this function are?

This is MCQ of a competitive exam(GATE), Answer is (d) given by GATE , and from other sources ,explanation is (b) somewhere and (d) somewhere , I am going with (b) as minimum at $270$, I have drawn ...
0
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0answers
27 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
0
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0answers
23 views

Hieroglyphic from Herschel to Babbage?

John Herschel sent a letter to Charles Babbage in which he included this hieroglyphic with the message "Interpret it, it contains a great discovery". Personally I have no clue what it could mean. ...
0
votes
1answer
25 views

Finding the functions for circular reflection and their inverted forms

I'm trying to solve this exercise: Show that the transformation of inversion in the unit circle is given analytically by the equations $$x'=\frac{x}{x^2+y^2}, y'=\frac{y}{x^2+y^2}$$Find the ...
1
vote
1answer
49 views

Example of $f:\mathbb{R}\to\mathbb{R}$ injective and bounded, but with inverse not bounded or injective.

I am trying to come up with an example of a bounded and injective function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}$ is not injective or bounded. What are examples that could apply in this ...
7
votes
5answers
402 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
2
votes
0answers
54 views

Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
0
votes
2answers
49 views

Prove that an equation has solution in R

Let $f:\mathbb R\to \mathbb R$, $x\in\mathbb R$ and $$f(x^2 + 3x + 1) = f^2(x) + 3f(x) + 1.$$ Prove that $f(x)=x$ has a solution $\in \mathbb R.$
0
votes
1answer
30 views

Non-linear system with functions

$f:\mathbb R\to \mathbb R$ monotonically increasing. Solve the system: $$\begin{cases} f(x) + x = f(y) + y\\ x^2 + xy + y^2 = 12\end{cases}$$ Since $f$ is monotonically increasing and ...
0
votes
1answer
21 views

Can functions within a matrix adjust its size?

I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix: ...
0
votes
1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
votes
1answer
48 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
0
votes
0answers
21 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
-2
votes
3answers
32 views

How to Prove it is a Even function? [on hold]

Let $y=f(x)$ be a function such that for any real numbers $a$ and $b$, $f(a+b)+f(a-b)=2[f(a)+f(b)]$ Prove that $f(x)$ is an even function.
0
votes
2answers
55 views

function: bending the y=x line

My question has many relative questions but I didn't find anything exact to my needs. Let's take the function $f(x)=x$ with $x\in[0,100] $. I need to bend this and make it a curve. f will be a ...
1
vote
1answer
35 views

Show that the angle between $OP$ and the normal to the curve at $P$ satisfies the following

I'm struggling to answer the following question below I've already worked out the gradient to the curve at $P$, but I'm having difficulty answering the second part of the question. MY attempt is as ...
2
votes
0answers
46 views

How do you the roots of functions that are not quadratics?

I was asked to consider the equation $(x-3)(x+3)^2=c$ I have been asked to find the values of C in which the equation has: three distinct roots only one real root a double root and a single root ...
0
votes
1answer
67 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
0
votes
1answer
75 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\mathrm{d}t$

Suppose $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, \mathrm{d}t$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I ...
-3
votes
4answers
78 views

Evaluate $ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$ [on hold]

How to evaluate limit of the following function at x=0 ? $$ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} } $$
1
vote
5answers
69 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
6
votes
5answers
94 views

Show that $f(x) = \log(x + \sqrt {x^2+1})$ is an odd function

I need to show that $f(x) = \log(x + \sqrt{x^2+1})$ is an odd function and from what I can understand from this question (found while searching): What is an odd function?, I have to show ...
0
votes
1answer
39 views

Prime Number Algorithm

function isPrime(n) { // If n is less than 2 or not an integer then by definition cannot be prime. if (n < 2) {return false} if (n != Math.round(n)) {return false} // Now assume that n is ...
0
votes
2answers
29 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
0
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0answers
15 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
0
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0answers
26 views

Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
0
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0answers
23 views

Functon fitting goes wrong

Let's say I got a some function (let's say it named $B_w$) and I make a curve deped on some parameters. As example ...
0
votes
2answers
41 views

Maximum value of $f(x)=\frac{x^2}{x^3+200}$ over natural numbers

This was a great problem I came across today.Just wanted to share it :-) $f$ is a function defined over the set of natural numbers(I mean the domain is natural numbers) by ...
5
votes
4answers
71 views

Asymptotic of Inverse Function

Suppose we choose a positive constant $c$ and let $f_c(x)=\frac12x^2+cx^{3/2}$. I would like to get an asymptotic estimate for the function $f_c^{-1}(x)$ as $x\rightarrow\infty$. I assume it will be ...
5
votes
1answer
57 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
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0answers
33 views
17
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4answers
2k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
0
votes
1answer
34 views

Approximate function as $x$ tends to infinity

I'm looking for a way to approximate the following function $f$ as $x \to \infty$ $$ f = \ln \left( 1 + e^{a_1 x} + e^{a_2 x} + A e^{(a_1+a_2) x} \right) $$ where $a_1$, $a_2$ and $A$ are constants. ...
0
votes
3answers
45 views

Why ordered sequences can be reduced to sets?

I am trying to understand why ordered sequences can be reduced to basic sets. I understand most of the following proof: Sequences can be defined as functions Functions are a special case of ...
0
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2answers
30 views

Drawing squares of functions

If I want to sketch the square of the sinc function, or any function for that matter, is there a neat transformation technique which would allow one not to refer to graphing devices for this task?
0
votes
3answers
49 views

Find the maximum of a non-linear function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=$$$$(-2(x+5)^2+200x)+(-2(y+10)^2+200y)+(-2(z+15)^2+200z)+(-2t^2+200t)$$ with $x+y+z+t = 150$ I don't know if ...
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0answers
30 views

Algebra 2 help!!?… [closed]

For the function below, find the difference quotient f(x)-f(a)/x-a and then simplify. f(x) =6x-5
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vote
3answers
67 views

Pigeonhole principle 3

I need help on this question, I'm lost and really don't know how to proceed: Use the pigeonhole principle to prove that in a round-robin chess tournament (with 18 participants) there will be at least ...
0
votes
2answers
25 views

Find the maximum of a function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=(2x+2)+(4y-1)+(3z+4)+(5t+3)$$ with $x+y+z+t = 50$ I don't know if this is feasible and how to proceed. I have ...
2
votes
1answer
35 views

General formula for a specific problem?

I have a problem which I would like to have a general formula for. Here is the description. There are island aligned by a grid. Every cell contains an island. Every adjacent island is connected by ...
0
votes
1answer
22 views

Find functions that satisfy this equation.

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant. [Edit]: with $n\gt{0}$
0
votes
1answer
48 views

Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...