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

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-1
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3answers
49 views

Can you help me prove that this function increases? [closed]

Can you help me prove that $f(x) = x/\ln (\ln x)$ is increasing on $(e^2, +\infty)$?
0
votes
1answer
27 views

solve system inequalities derived from a function

I have this system of inequalities $$ \begin{cases} y^2-3 \geq 0\\ 16y^4-96y^2 \geq 0 \end{cases} $$ the solution for the first inequality is $y\leq -\sqrt{3}$ or $y\geq \sqrt{3}$ and the solution ...
4
votes
2answers
53 views

Functions and Derivatives

Generaly curious: Let there be a set of functions: Will the sum of the derivatives of the functions be equal to the derivative of the sums?
1
vote
3answers
96 views

Find the range of $y = \sqrt{x} + \sqrt{3 -x}$

I have the function $y = \sqrt{x} + \sqrt{3 -x}$. The range in wolfram is $y \in\mathbb R: \sqrt{3} \leq y \leq \sqrt{6}$ (solution after correction of @mathlove) $\sqrt{x} + \sqrt{3 -x} = y$ $$ ...
1
vote
2answers
23 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
1
vote
5answers
34 views

Give an example of a function from A to B that is not one-to-one. Explain why it is not one-to-one

A= {a,b,c,d} B= {1,2,3,4,5} Currently studying for a final. I know that a one-to-one function cannot map to 2 elements. There are more elements in B than in A. I don't know how to give a specific ...
2
votes
0answers
37 views

Why does the definition of a bounded function, and its negation, seem to prove contradictions?

If I want to show that $f(x) = \dfrac{1}{x}$ is not bounded above on $(0,1]$, the way I have learnt to proceed is to assume that $f$ is bounded above, so by definition there is a number $M > 0$ ...
0
votes
1answer
9 views

Find a Weight Function with specific characteristics

I need to build a weight function and I want to understand how you would do that. The reasoning you would use to define it. My function has to be something like: $f(\alpha)$ which is: $0$ if ...
-1
votes
1answer
30 views

area and volume [closed]

The total length of all 12 sides of a rectangular box is 60. (i) Write the possible values of the volume of the box. Your answer should be an interval. Now suppose in addition that the surface area of ...
1
vote
0answers
27 views

Function that satisfies the given (x,y) values

I am trying to come up with a function that (approximately) satisfies these (x,y) values. (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 6), (9, 5), (10, 4), (11, 3), (12, 2), (13, 1), (14, 2), ...
6
votes
2answers
72 views

Conjecture function $g(x)$ is even function?

Let $f,g:R\to R\setminus\{0\}$ and $\forall x,y\in R$,such $$\color{crimson}{f(x-y)=f(x)g(y)-f(y)g(x)}$$ I have prove the function $\color{crimson}f$ odd function. because let $y=0$ we have ...
0
votes
3answers
30 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let ...
2
votes
1answer
54 views

How do we know which terms are of higher order?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] At this point, we have shown ...
0
votes
1answer
35 views

How to Integrate by Parts when One Function Is a Polynomial?

So I have the following integration: ∫(x -2)(cosx)dx between the intervals (-a,a). we know that, The integral of a even function over the range (-a,a) can be rewritten as twice of the intergral ...
2
votes
1answer
37 views

When is a balance assumption consistent?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] The final case to try is to ...
0
votes
0answers
17 views

Is there a word for two functions that can be defined using each other?

Let's say I have two functions f and g. For example: f(x) = x + 1 g(x) = x - 1 And I can use function g to define function f just be somehow manipulating the ...
0
votes
1answer
50 views

How to prove that $x\csc x <\pi/3$

If $x \in (0,\frac \pi6)$, then using calculus prove that $x\csc x<\frac \pi3$ My attempt: let $f(x)=\csc x$$$\implies f'(x)=-\frac{\cos{x}}{\sin^2 x}$$ which is less than $0$ for all $x\in ...
1
vote
1answer
25 views

Preimage of $[0,1]$ under $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}$

I want to find $$ f^{-1}([0,1]) $$ where $$f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}.$$ I have to do this in order to find a dessins d'enfant associated ...
0
votes
1answer
28 views

Find the function the range of the $\frac{5+2\sqrt{3+2x}-x}{\sqrt{x+1}+\sqrt{3-x}}$

Find the function range $$f(x)=\dfrac{5+2\sqrt{3+2x}-x}{\sqrt{x+1}+\sqrt{3-x}}$$ since $$\begin{cases} 3+2x\ge 0\\ x+1\ge 0\\ 3-x\ge 0 \end{cases}$$ then the function domian is $-1\le x\le 3$ and ...
-2
votes
2answers
31 views

Area of polygon inscribed in a circle [closed]

Let $A_n =$ the area of a regular $n$-sided polygon inscribed in a circle of radius $1$ (i.e., vertices of this regular $n$-sided polygon lie on a circle of radius $1$). ($i$) Find $A_{12}$. ...
0
votes
1answer
12 views

Is there a proper term for sets of interdependant functions?

I am looking for a term that would describe the sets of functions that are very closely related, such as: the trigonometric functions $\sin$ and $\cos$ the hyperbolic functions $\sinh$ and $\cosh$ ...
2
votes
1answer
56 views

Show that $\int_{0}^{1}P(x)f(x)=0$ [closed]

Let $f:[0,1] \rightarrow \mathbb{R}$ a continuous function, such that $\exists (u,v)\in [0,1]$; $ f(u)>0, f(v)<0$. Show that there exists $P$ a polynomial such that $P>0$ over $[0,1]$ and ...
1
vote
4answers
30 views

On proving the surjectivity of non-injective functions.

As given in my lectures and several other areas, the definition of a surjective function is "a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y has ...
2
votes
1answer
29 views

Maximizing sum of logarithms (Z-channel capacity)

In the context of information theory, I am trying to maximize the following function (mutual information of the Z-channel's input and output) with respect to $p$ in order to derive Z-channel's ...
0
votes
0answers
25 views

A certain question about elementary logic and functions.

I need to understand the following detail. Suppose we know the following is true(we know that composition for functions $\theta, \sigma$ and $\tau$ is well-defined): There is a unque isomorphism ...
0
votes
3answers
20 views

Inverse of a function on two sets.

I understand that $f^{-1}(A\cup B) = f^{-1}(A)\cup f^{-1}(B)$, but what is $f^{-1}(A\cap B)$? Is it necessarily $f^{-1}(A)\cap f^{-1}(B)$?
0
votes
3answers
63 views

Use / Don't use Rolle's Theorem

I've got an interesting exercise and I tried to use Rolle's Theorem to prove it. Do you think my prove is good or what should I use to prove it? Exercise : For continuous function $f$ we've got : ...
0
votes
3answers
20 views

Differentiation of subtraction

I've got an exercise to do and I don't really know what to do. Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist ...
0
votes
2answers
27 views

If $g(x) = 2f(x) + 5$, find the value of $g^{-1}(x)$ [inverse]. Considering $f(x)$ is invertible

I know that when an invertible function is inverted, the domain becomes the range and viceversa. That implies, Value of g(inverse) = Domain of g(x) = 2(Range of f(x)) +5 So my answer would be ...
3
votes
3answers
79 views

Is there a bijective function $f: \mathbb{R}\to\mathbb{R}$ that is discontinuous?

Is there a bijective function that is discontinuous?
1
vote
1answer
19 views

Modular zero of a function

Is there a quick and dirty way to find the modular zero of a function, such that f(x) = 0 (mod p), p being a prime. E.g. f(x) = x^2 - 5x + 1 and p = 7. It's quite easy to find that for x = 6 the ...
2
votes
1answer
29 views

Continuous function on compact interval $[a,b]$ with non-negative values

let $f:[a,b]\longrightarrow[0, \infty)$ be a continuous function satisfying the following: $f(\frac{a+x}{2})+f(\frac{2b+a-x}{2})=f(x), \forall x \in [a,b]$. Then the only function that satisfies these ...
1
vote
1answer
20 views

How to notate the restriction of an inverse of a function?

Let $f$ be a function, which is defined on $E\subset\mathbb{R}^n$, and which is mapping to an extended real numbers. That is, $$f:E\longrightarrow\overline{\mathbb{R}}$$ Then, for a subset $A$ of $E$ ...
2
votes
2answers
71 views

How can I show $1-\frac{1}{x}+x^{1-\frac{1}{x}}<x$ for real $x>1$?

Denote $$f(x):=1-\frac{1}{x}+x^{1-\frac{1}{x}}$$ How can I prove that $f(x)<x$ holds for every real $x>1$ ? Wolfram gives the taylor series ...
1
vote
1answer
51 views

Find $f$, such that $e^{f'(x)}+f(x)=e^{-x^2}+e^{-2xe^{-x^2}},\ \forall x\in\mathbb{R}$

I have constructed a great exercise but now I don't know how to solve it without using sequences. My original thought was kind of "EVT in an open interval" with reductio ad absurdum. Can you help me ...
0
votes
0answers
31 views

How can show the following function is log-concave?

Suppose that $g(x)$ is an increasing function and $0\leq g(x)\leq1$. I was working on a problem and it reduced to show that if $1-g(x)$ is log-concave then $$f(x)=(1-g^a(x))^b, a\geq 1, b,x>0$$ is ...
0
votes
0answers
18 views

specific convergent functions question

Can someone help me formulate a direct proof of: If g(x) does not equal 0 for all x in the Domain and the limit of g(x) as x approaches x0 does not equal 0, then f/g has a limit at x0 and limit as x ...
0
votes
2answers
24 views

Can $y$ be made the subject in such equation?

I came across the equation: $${x^2} + 2xy + {3y^2} = 1$$ I tried making $\space y \space$ the subject, but the furthest I got was: $$ 2xy + {3y^2} = 1 - {x^2} $$ $$ y(2x + 3y) = 1 - {x^2} $$ $$ y ...
0
votes
0answers
29 views

Lebesgue measurable function equals Borel measurable function a.e.

I am trying to prove that if $f: \mathbb R^n \to \overline{\mathbb R}$ is measurable, then there exists $g: \mathbb R^n \to \overline{\mathbb R}$ Borel measurable such that $f=g$ a.e. I know this ...
1
vote
1answer
51 views

Functional equation - nonlinear

How do I go about solving this functional equation? $ 2f(n) = 2n^2 + f(2n) $, where f(n) takes integervalues.
3
votes
1answer
48 views

Given bijection between $\mathbb{N}$ and $A$ and $B$, find bijection from $\mathbb{N}$ to $A \cup B$

Let $A$ and $B$ be two countable sets and consider that $f$ is a bijection from $\mathbb{N}$ to $A$ and $g$ is a bijection from $\mathbb{N}$ to $B$. I have to find a bijection from $\mathbb{N}$ to $A ...
0
votes
3answers
46 views

Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
1
vote
1answer
75 views

Problem involving polynomial function and prime numbers

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
1
vote
1answer
43 views

Difference between two zero-one indicator functions

I have a zero-one indicator function $I(cond)$ which returns $1$ if the condition cond is true and $0$ if cond is false. Now I have the following difference: $I(a = c) - I(b = c)$. For some reason, I ...
0
votes
1answer
102 views

Find a positive convex function $f$ defined on $[a,b]$, s.t. $f(a)\times f(b)=1$ and $\int_a^b{f'^2dt}=12$

Find a function $f:[a,b]\to \mathbb{R}$ which is convex on $[a,b]$ such that $\int_a^b{f(t)dt}=0$, $\int_a^b{f'^2(t)dt}=\frac{12}{b-a}$, and $f(a)f(b)=1$? Another similar question which states: Find ...
0
votes
1answer
26 views

Finding a function with given partial derivatives dx dy

I need to find a function $f(x,y)$ such that $f(x,y)dx = \frac{1}{2}\frac{x}{\sqrt{x+y}}$ and $f(x,y)dy = \frac{1}{2}\frac{y}{\sqrt{x+y}}$ how can this be solved?
3
votes
2answers
83 views

What is implied by $f \circ g = g \circ f$?

For any two functions $f(x)$ and $g(x)$ we are given $f \circ g = g \circ f$. What does this imply? I found that $f(x) = g(x)$, $f(x) = g^{-1}(x)$ and $ f(x) = x \ (\neq g(x))$ are some of the ...
0
votes
0answers
23 views

Topology: every continuous function on $\mathbb{R}^2$ scales a point

The question is simple: Suppose $f : \mathbb{R}^ 2 \to \mathbb{R}^ 2$ is continuous. Show that there exist $\lambda > 0$ and $x \in \mathbb{R}^2$ such that $f(x) = \lambda x$. So basically, we ...
3
votes
2answers
33 views

is $y = \sqrt{x^2 + 1}− x$ a injective (one-to-one) function?

I have a function $y = \sqrt{x^2 + 1}− x$ and I need to prove if it's a Injective function (one-to-one). The function f is injective if and only if for all a and b in A, if f(a) = f(b), then a = b ...
0
votes
0answers
13 views

generate unpredictable value with a shared key

I will start with an example. Some beacon like estimote Beacons use something called secure ID. The ID transmitted by the beacon change every 10 minutes autonomously without contacting the server. ...