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

learn more… | top users | synonyms (1)

0
votes
1answer
37 views

Application of Inverse Function Theorem

This is a seemingly easy exercise. Yet I am not sure if I am missing any finer details here as this is listed as one of the challenging problems on Dr. Epstein's (Upenn) course site for real analysis. ...
2
votes
1answer
34 views

How can we apply the definition?

Show that $$g(x, y)=ye^x+\sin x+(xy)^4$$ is continuous. The definition is: $f : A \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $x_0 \in A$ iff $\forall \epsilon \exists \delta:$ ...
3
votes
2answers
57 views

Functions and Sets

Robert, Susan, and Thomas are the sole contestants in a lottery in which two prizes will be awarded. Three tickets with their names on them are placed in a hat. The person whose name is on the first ...
1
vote
3answers
50 views

Total number of functions $f\colon S\to S$ where $S=\{1,2,3,4\}$

I missed a lecture on this topic and I'm having a hard time figuring out how this discrete function works. I'm given $S=\{1,2,3,4\}$ and $F =$ all functions from $S$ to $S$. What does this mean? I ...
0
votes
1answer
23 views

What function can produce a perfect saddleback plot and fulfil the following requirement?

I need to find a function that produce a good saddleback plot. The function has the following requirements: Having 2 arguments: x and y Both x and y are natural numbers The result of the function ...
0
votes
1answer
12 views

finding the product of sides in a traingle using function with 2 variables

ABC is a triangle,M is a variable point inside it. Let AB=c,CA=b,BC=a. Let x,y,z and alpha be the respective areas of the triangles MBC,MCA,MAB,and ABC. Let I,J,and K be respectively the orthogonal ...
0
votes
0answers
28 views

How should prove that a function is onto?

This is what is done to prove that a function is onto: For functions given by formulas we proceed along the following lines. Step 1: Let y be any element of the codomain and x an element of the ...
0
votes
0answers
19 views

How to analyze the convexity of the this function? Or how to analyse this function in general?

For some $n\in\mathbb{N}$, I have a function $f_i(\mathrm{x})$ for $i\in\{1, \ldots, n\}$ of the form: $$ f_i(\mathrm{x})=f_i(x_1, \ldots, x_{i-1}, x_i, x_{i+1},\ldots, x_n) = ...
0
votes
0answers
24 views

Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
0
votes
1answer
21 views

how to determine the lowest number in this curve

can someone help me on how to find the lowest value which the function will get ? $y=\frac{1}{2}(e^x-e^{-x})+\frac{1}{2}n(e^x+e^{-x})$ thought of using t as $e^x$..but couldn't get an answer.
0
votes
4answers
50 views

Prove that f is either strictly increasing or decreasing

Let $f: [0,1]\to [0,1]$ be continuous and one-to-one. Prove that $f$ is either strictly increasing or strictly decreasing. Sorry if this is a duplicate question. Not sure whether or not to prove this ...
0
votes
0answers
7 views

Function to increase entropy for a specific number and seed and reduce it for the rest

Hello I think I am wording the title correctly. I am looking for a function / algorithm that can increase the variability or entropy of a specific number and reducing it for the rest. The function can ...
0
votes
1answer
30 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
-1
votes
3answers
91 views

How do you solve $f(x)=4$ for $x=2$ [closed]

If I have the following problem $a(x)=4$ For $x=2$ How do you solve it? If I look at $a(x)$ as function it will be $a$ of $2$: $a(2)= 4$. If I look at $a(x)$ as just variables it will be $a(2)=4 ...
6
votes
4answers
60 views

$f(A \cap B)\subset f(A)\cap f(B)$, and otherwise?

I got a serious doubt ahead the question Be $f:X\longrightarrow Y$ a function. If $A,B\subset X$, show that $f(A \cap B)\subset f(A)\cap f(B)$ I did as follows $$\forall\;y\in f(A\cap ...
3
votes
1answer
13 views

How to determine a function?

I have two relations: $R_1:=\{(x,y)| x^2=y\} \subseteq \mathbb{N} \times \mathbb{R}$ $R_2:=\{(y,x)| x^2=y\} \subseteq \mathbb{R} \times \mathbb{N}$ $R_1$ is a total function $R_2$ is a partial ...
0
votes
0answers
23 views

Concave functions inequalities

If we have a differentiable concave function $u(x)$ defined on $p$ and $x$, such that $u(x_1) > u(x_2)$, Does the concavity and differentiability of the function in any way imply that: $u(x_1) ...
1
vote
1answer
35 views

Convexity of Conjugate Function on Determinant

If $f: \mathbb R^n \to \mathbb R$ then $conj(f) : \mathbb R^n\to \mathbb R$ is defined as $conj(f)(y) = \sup_{x \in dom_f} (y^Tx - f(x))$ and it is called the conjugate function of $f$. if $f(X) = ...
1
vote
6answers
77 views

Functions $\mathbb{N} \to \mathbb{N}$ that are injective but not surjective, and vice versa

Suppose that $A$ and $B$ are sets each containing the same finite number of elements and $f: A \to B$. a. Prove that $f$ is injective if and only if it is surjective. b. Give an example ...
3
votes
2answers
42 views

Function with an $x$ not in simple form

I've stumbled upon a practice example in an old textbook which I find confusing. Maybe it's because I haven't reached part of an explanation yet (went through pages, haven't found anything of help). ...
1
vote
2answers
56 views

Finding an irrational function with horizontal asymptotes y=1 and y=5

I can't seem to find an irrational function with $2$ horizontal asymptotes $y=1$ and $y=5$. I've looked everywhere and tried all I know, I keep getting $2$ asymptotes that the contrary of each other ...
3
votes
1answer
25 views

Constructing a recursive definition.

I know a recursive definition is a function or procedure that is defined in terms of itself, for instance $f(n) = f(n - 1) + n$ or $f(n + 1) = f(n) + n + 1$. This makes sense to me in terms of ...
0
votes
1answer
26 views

Write this piece wise function in terms of the unit step function unit step

$$f(t)=\left\{\begin{array}{rl} 1, & 0\leqslant t <1\\ -2, & 1\leqslant t\leqslant2 \\ 0, & t>2 \end{array}\right.$$ I got a long drawn out answer that can't ...
0
votes
1answer
33 views

function to approximate $x!$ without factorial

I am looking for a function $f(x)$ such that $f(x)\approx x!$, but (obviously) the function of x does not use factorial, eg a polynomial or exponential function. it does not have to be precise, just ...
0
votes
2answers
47 views

Name of Property of Functions

Is there a specific term to describe the following property of a function: $$f(ab)=f(a)f(b)$$ And if $f(x)$ satisfies this property and is not the trivial functions $x$ or $0$, must $f$ be ...
2
votes
2answers
92 views

How can I interpret the ratio $\frac{f(x_0)}{f'(x_0)}$?

let $f'(x)$ the first derivative of function $f(x)$. For some $x_0$, how can I interpret the ratio $\frac{f(x_0)}{f'(x_0)}$ ? More specifically, what does it mean a $\frac{f(x_0)}{f'(x_0)} \gg 1$ ?
-1
votes
1answer
40 views

What are the domain and range of the function graphed below? [closed]

I don't know what do with this question please help!!!! it's a online question.
0
votes
3answers
46 views

Can functions return sets?

The definition of a function is (link): A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set ...
0
votes
0answers
13 views

Formally correct “generator expression” for parameters of a function

I'm trying to express formally correct that a class of functions exists that have a certain property that applies to all concrete "instances" of this class. In that I try to write a "generator ...
1
vote
2answers
31 views

Function needed to fit data

I ran some computations and there seems to be some neat relationship between my variable $t$ and the corresponding result. What function should I try first to fit my data? It is given that: $t$ is ...
1
vote
1answer
50 views

Under what condition(s) we can use the following relation: $\frac{df(x)}{du}=\frac{df(x)}{dx}\frac{dx}{du}$

Let $\frac{df(x)}{dx}$ the first derivative of function $f(x)$ and we assume that $x=x(u)$ (i.e. depends on $u$). Under what condition(s) we can use the following relation: ...
2
votes
3answers
17 views

Onto and One to one functions given composite is also onto or one to one

if $f:X→Y$ and $g:Y→Z$ are functions and $g∘f$ is one to one, are $g$ and $f$ also one to one? Similarly, Are they also onto? How can i prove these or disprove these with examples?
-4
votes
3answers
29 views

One to one functions - If $g \circ f $ is one to one, prove that $g$ is not one to one [closed]

How can I I prove that if $g \circ f$ is one to one, then $g$ is not one to one? Whats an example of it not being a one to one function? Similarly, what about f not being onto if $g \circ f$ is onto ...
0
votes
2answers
34 views

How to modify this bump function so that the “bump” is at $y=1$?

$$f(x) = \begin{cases} e^{-1/(1 - x^2)} & -1 < x < 1\\ 0 & \text{otherwise} \end{cases} $$ I noticed that when I multiply the denominator of the fractional part of this ...
4
votes
1answer
52 views

Are these equivalent? $\cos^2(5x) = (\cos(5x))^2$

Are these equivalent? $$\cos^2(5x) = (\cos(5x))^2$$
0
votes
1answer
17 views

Solving exponential equation - Order of operations

Hopefully this should be a quick questions. When solving the exponential equation 5 * 2^(u/2) + 30 = 600 Why do you subtract 30 first and not divide 600 by 5? The order of operations indicates that ...
1
vote
1answer
41 views

Why do I need to show uniqueness?

Question Let $c$ be a cluster point of $A \subset \mathbb{R}$, and $f:A \to \mathbb{R}$ be a function. Suppose for every sequence $\{ x_n \}$ in $A$, such that $\lim_{n \to \infty} x_n = c$, the ...
1
vote
2answers
25 views

Show that $\frac{1}{x}-\sin(x)$ has exactly one root in the interval $(0,\frac{\pi}{2}]$

I have problems showing, that this function has exact one root in the interval $\left(0,\frac{\pi}{2}\right]$: $$f(x):=\frac{1}{x}-\sin(x)$$ My idea was to use the Intermediate value theorem, but ...
1
vote
1answer
14 views

Order of operations when balancing equation

This is a trivial question, yet I have only really thought about it today and would like some insight. To find the $x$-intercept of the following function: $$f(x) = 2 \sin x + 1$$ We set $f(x) = ...
1
vote
3answers
59 views

Is $f(x) = (ax+1)^b$ a power law?

Pardon my ignorance, but is it appropriate to call the following function a "power law"? $f(x) = (ax+1)^b, x \ge 0$ (Update) It is given that $b \le 0$.
0
votes
2answers
41 views

How to construct a diffeomorphic function using another function with certain properties

A $C^{\infty}$ function $f(x)$ on the interval $[a, b]$ satisfies the following 3 properties: 1) $f(x) = 1$ for $a \leq x \leq b$ 2) $f(x) = 0$ for $x < \alpha$ and $x > \beta$ where $\alpha ...
0
votes
1answer
20 views

How to modify a function to meet certain properties?

I want to modify $$B(x) = \left\{ \begin{array}{lr} e^{-\frac{1}{x^2}} & : x > 0\\ 0 & : x \leq 0 \end{array} \right.$$ so that the new function $$C(x) = \left\{ ...
1
vote
2answers
20 views

If f(x, y) is convex, is g(x)=f(x, c) convex, for any constant c?

If $f(x, y)$ is convex (concave) defined on $\mathbb{R}^2$ and $g(x)=f(x, c)$, $c\in \mathbb{R}$, then is $g(x)$ necessarily convex (concave)?
3
votes
3answers
41 views

$f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x) $

$f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x)$ Determine the greatest and least values of $\frac{39}{f(x)+14}$ and state a value of x at which greatest values occurs. Do I just use a graphing ...
0
votes
1answer
25 views

Pdf of a linear transformation $f(x)=1-|x| ,-1<x<1$. Find pdf of $Y=X^2$. [closed]

Let $X$ be a continuous random variable with density function $$f(x)=\begin{cases} 1 - |x| &\text{if }-1<x<1,\\ 0 &\text{otherwise}\end{cases}.$$ Find the density function of $Y = X^2$. ...
1
vote
0answers
22 views

Calculate $\frac{d}{dt}\inf_{x=a+b}(a+bt)$.

I'm trying to calculate the derivates with respect to $t≥0$ (real) for this function $$f(t)=\inf_{x=a+b}(a+bt)$$ where x, a, b are fixed positive real numbers. I got $$f(t+\Delta ...
2
votes
1answer
51 views

Can a function $f:\mathbb{R^n} \mapsto \mathbb{R^m}$ where $n<m$ be surjective?

I have thought about the following problem: We have a function $f$, which has as the domain $A \subseteq \mathbb{R^n}$ and maps its inputs to a higher dimensional space $B \subseteq \mathbb{R^m}$ ...
1
vote
1answer
49 views

Ring of Infinitely Differentiable Functions

Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$. I have managed to prove that this ring is a Principal Ideal Domain. I must also prove ...
1
vote
0answers
25 views

What function should I use to fit this data? Logistic function is not good enough fit.

I'm using curve_fit from numpy to try an fit some data I have. The function I used was the logistic function defined http://en.wikipedia.org/wiki/Logistic_function and the parameters I picked were ...
1
vote
1answer
24 views

finding the local extremum of a function of 2 variables

consider the function with 2 variables $f(x,y)=-2(x-y)^2+x^4+y^4$ (n.b.: f is from R^2 onto R). I proved that it has 2 local minimums other than f(0,0) but how can i prove that f(0,0) is a local ...