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

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0
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1answer
16 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
48 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 ...
6
votes
5answers
232 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
32 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
45 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
27 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
47 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
47 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
53 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
33 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
66 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
91 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
28 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
votes
0answers
14 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
votes
0answers
25 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
votes
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
40 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
68 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 ?
0
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0answers
33 views
16
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4answers
1k 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
41 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
votes
2answers
29 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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votes
0answers
29 views

Algebra 2 help!!?… [on hold]

For the function below, find the difference quotient f(x)-f(a)/x-a and then simplify. f(x) =6x-5
1
vote
3answers
63 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
34 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 ...
0
votes
2answers
33 views

Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from ...
1
vote
2answers
54 views

Why is there no inverse function in this case?

There is no inverse function for $R(q) = 40q - 4q^2$ for $0≤q≤10$ But when $0≤q≤5$ there is inverse function. It's something about the function being one-to-one but I don't know why it isn't ...
1
vote
1answer
35 views

Is this relation symmetric and transitive?

Set A is given as $A = \{1,2,3,4,5,6,7,8,9,10,11,13,14\} $ And is defined as $R = \{(x,y) : 3x = y\}$ The relation that I'm getting is: $ R = \{(3,3), (6,6), (9,9), (12,12)\} $ Over here, it is ...
0
votes
1answer
8 views

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
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votes
0answers
11 views

Constructing a function [on hold]

The revenue R that a store earns from selling a particular item is given by the product of the number of units sold x, and the price per unit p. Construct the revenue function for the price function ...
0
votes
2answers
34 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
-4
votes
2answers
60 views

How to simplify $(x+1) / (x^3-x)$ [on hold]

For all $x$ in the domain of the function $\frac{x+1}{x^3-x}$, this function is equivalent to which of the following? (A) $\dfrac{1}{x^2}-\dfrac{1}{x^3}$ (B) ...
0
votes
1answer
15 views

how many matches are played

so the question is the are a total of 16 teams. Each team will play each other once so each team will play 15 matches since they can't play themselves. How many matches in total are played?
0
votes
3answers
95 views

$f(x)$ is a periodic function. What is its period?

Suppose that $f(x)$ is a periodic function. If we have: $$\forall x :f(x+346)=\frac{1+f(x)}{1-f(x)}$$ What is its minimum period?
0
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3answers
38 views

How to find graph of the sum of two functions

Suppose I know the graphs of two functions $f(x)$ and $g(x)$. How can I find the graph of $h(x)=f(x)+g(x)$? What are the rules to be followed ? P.S. In case my question seems silly,at least provide ...
0
votes
2answers
53 views

A question about Idempotent functions [closed]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...