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

learn more… | top users | synonyms (1)

0
votes
1answer
14 views

function - Discrete Math

How can i Declare and define this statement : the variable n_pair which is a pair of positive even numbers, the first of which is at least as great as the second. How should I approach this !?
1
vote
1answer
30 views

Existence of functions $g$ such that 1. $f\circ g(1) =2$; 2. $g \circ f(1) = 2$, for all $f$ [on hold]

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$. a) Prove or disprove the statement: "For all $f \in F$, there exists $g \in F$ so that $(f \circ g)(1) = 2$" b) Prove or ...
1
vote
1answer
20 views

Absolute Maximum and Minimum of cos function

I am having a little trouble trying to figure out the following problem: Find the absolute maximum and minimum values of the function $f(x) = x-2\cos x$ on the interval $[0, 2\pi]$. I have taken the ...
0
votes
0answers
14 views

Find function $f(x,y)$ such that $f(x,0) = J(x)$ and $\nabla_{(x,y)} f(x,y) = g'(y)h(J,\nabla_x J)$?

Let $J:\Omega \to \mathbb{R}$ be a smooth function such that $0 < C_1 \leq J(x) \leq C_2 < \infty$. Is it possible to find a function $f:\Omega \times [0,\infty)$ such that $$f(x,0) = J(x)$$ ...
0
votes
1answer
34 views

real analysis -functional equation

Be $f:\mathbb R^ +\mapsto\mathbb R$ a function that satisfies the following conditions: a)$ f(f(f(x)))+2x=f(3x)$ for every $x\gt 0$; b) $\lim_{x \to \infty} (f(x)-x)=0$. This was proposed by ...
0
votes
0answers
17 views

Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
0
votes
1answer
19 views

Determine if a function is even or odd

Let $f:\mathbb{R}\to\mathbb{R}$. Define $h:\mathbb{R}\to\mathbb{R}$ by $$h(x)=f(x)\{f(x)+f(-x)\}$$ Then, which of the following option(s) is/are correct ? (A) h is even for all f (B) h is odd for ...
-3
votes
1answer
31 views

How can I get a number between $0$ and $1$, that will be larger as another number is larger?

I have a number $x$, ranging roughly between $0$ and $35000$. I want to create a number $y$, which will be in the range between $0$ and $1$. $y$ will be larger when $x$ is large and smaller when $x$ ...
0
votes
1answer
43 views

Solving several equations involving sine function

Recently, I asked for root of this equation $2x - \sin(2x) = \frac{\pi}{2}$, then i got $x = \frac{Dottie}{2} + \frac{\pi}{4}$. Thanks everyone. Now can i define a function like this: $f(n) = x$ to ...
5
votes
1answer
30 views

Maximum value of function given minimum value

Suppose there is a function $f(x)=\frac{x^2-2x+b}{x^2+2x+b}$ (the problem doesn't specify, but I am assuming $b$ is a real) that has a minimum value of $\frac{1}{2}$. What is the maximum value of ...
3
votes
2answers
41 views

How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question. It's telling us to explain the changes that were made to the ...
1
vote
1answer
28 views

Logarithm function from graph (word problem)

I was hoping to find some hint to solve this: There is an internet company that wants to price their service $30/month, which will include 25GB on their package. After 25GB, the price will increase ...
2
votes
2answers
57 views

Is there a non-decreasing function that is discontinuous at every rational point? [duplicate]

A well-known theorem is that if $f:[a,b]\to\mathbb{R}$ is non-decreasing, then $f$ as at most countably many discontinuities. This led me think of the following question. Question: Is there a ...
0
votes
1answer
26 views

Is this problem/counterexample stated correctly/valid?

This problem was given by the teacher as a practice exercise. Is it valid? If $f:M \rightarrow N,$ $g:M \rightarrow P$, and $h: P\rightarrow N$ are maps with $g$ surjective and $h$ injective, show ...
0
votes
1answer
25 views

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$. I have difficulties with this one when i try to find the limits, i get indefinites that look like (infinity over infinity)over ...
1
vote
1answer
37 views

How do we know that the Maclaurin Series can always be used to approximate a function?

I'm aware of the formula that can be used to derive the Maclaurin series for a particular function: My question is - how do we know that all functions can be represented as an infinite series of ...
3
votes
2answers
48 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
1
vote
1answer
31 views

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at x=0 there should be a vertical asymtote, but i don't know how to formally execute this..

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at $x=0$ there should be a vertical asymtote , but i don't know how to formally execute this. When i try to find the limit, i get the square ...
1
vote
1answer
26 views

why is the domain of the function in the picture >=0, why is it not just !=0

http://i.imgur.com/Ofmw9ieh.jpg Expression - $$z=\sqrt[131]{\frac{x^2+y^2-2x}{2y-y^2-x^2}}$$ I do not understand why the expression under the odd root has to be greater than or equal to 0 according ...
0
votes
1answer
30 views

How would you methodically find a tangent/and normal line of a point T$\notin (x,f(x))$? [on hold]

How would you methodically find a tangent/and normal line of a point T$\notin (x,f(x))$ ? For example the normal/ and tangent lines from point (-1,-1) to $y=x^2$ . Just interested in 2 dimension ...
0
votes
0answers
27 views

Using Tikz package to draw piecewise functions [on hold]

I would like to use Tikz package in Latex to draw the following piecewise functions and their epigraphs: 1) $f:[0, \infty)\rightarrow \mathbb{R}$ such that $f(0)=1$, $f(x)=0$ if $x>0$; 2) $g:[0, ...
1
vote
0answers
52 views

Make $x$ the subject for $y=xe^x$ [on hold]

How do I make $x$ the subject for $y=xe^x$? Taking $\ln$ for both sides, I can't express it as $x = f(y)$.
0
votes
1answer
52 views

Simplifying identity cos*cos+sin*cos

$$\cos(3\pi/2 - a) = -\sin(a)$$ According to an answer to one of the questions in my book that's true, but come up with that? $$cos(\pi/2 - a) = \sin(a),$$ but this is $3\pi/2$. do you just ignore ...
1
vote
2answers
37 views

Showing a complex function is constant

If I know that a function $f$ is entire and $f(z)=f(z+1)=f(z+i)$ for all $z \in \mathbb{C}$, how do I show that $f(z)$ is constant? I feel like this needs use of the uniqueness/identity theorem to ...
1
vote
0answers
30 views

Proving existence and uniqueness of solutions to the functional equation $f(n) = r \cdot f(n-1)$

Suppose I have a functional equation $f(n) = r \cdot f(n-1)$ where $r$ is a constant. This represents a geometric progression and a known solution is $g(n) = ar^n$ where $a = g(0)$. By intuition, ...
1
vote
1answer
22 views

Function inverse

Question: Let $S$ and $T$ be sets and let $f:S\to T$. Show that $f$ is a surjection from $S$ to $T$ iff for each subset $B$ of $T$, $f[f^{-1}[B]]=B$. So it's a surjection if for each element $t$ ...
0
votes
0answers
9 views

Choosing a set of functions for linear combinations

I need to approximate some real valuated function $f(x)$ with a linear combination of functions from a set $\{f_i(x)\}$ to solve a practical problem. The domain is rectricted to $x$ in the range ...
1
vote
0answers
9 views

Internalizing results about composition and surjectivity/injectivity

I'm trying to see if there is any intuition pump / analogy that allows me to internalize ( and readily derive them) a series of results about the concepts of composition mixed with ...
-3
votes
2answers
76 views

Showing $\lim_{(x,y) \to (0,0)} \sin (xy) / xy = 1$

How could we show that $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\sin xy}{xy}=1$$ ?? Could you give me some hints ?? EDIT: Could we show it as followed?? $$\lim_{(x,y) \rightarrow (0,0)} ...
0
votes
0answers
24 views

polynomial function Differential equation - polynomial 1 [on hold]

$$(1-x^3)·y'-y^2+(x^2)·y+2·x=0$$ solve the differential equation , knowing that it can take as a solution a polynomial function (2nd grade) .
0
votes
0answers
10 views

Proof of Strictly Increasing Functions

For real numbers c, d with c$<$d we denote the open interval in R by (c,d)=x$\in$R: c$<$x$<$d. Recall that a function f:R-->R is strictly increasing if for all x,y in domain of f, whenever ...
-2
votes
1answer
14 views

Dirchilet Function with a Sequence [on hold]

If there is a Dirichilet function and a sequence an(which has a limit). Does D(an) might have a limit? thank you
0
votes
1answer
34 views

How to draw the graph of $x^6 = y^3$ and $3y = (log x)^2$?

For example, if the equation is $x^4 = y^2$ then I can separate this to get the two equations $y = x^2$ and $y = - x^2$ , hence I can plot the two graphs. But how do I simplify the given equation? I ...
1
vote
0answers
26 views

Functions, sets, intersections

Question: Let $S = \{1,2\} = T$ and define $f: S \to T$ by $f(1) = 1 = f(2)$. Let $B = \{1\}$ and let $C = \{ 2\}$. Find $f[B \cap C]$ and $f[B] \cap f[C]$ and observe they are not equal. Hint: ...
0
votes
2answers
42 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
1
vote
1answer
16 views

For which value of a , b , d these 2 functions are equal

Functions : $$ f(x) = (ax+b)/(x+d) $$ And the inverse one : $$ g(x) = (xd-b)/(a-x) $$ I tried to solve it and I got this : $$ a(-x^2 + ax + b) = d(x^2 + xd -b) $$ But I can't go further, How can I ...
0
votes
1answer
16 views

Has the functions having countably infinite image, but finite when the domain is bounded, a conventional name?

I'm trying to find properties for functions that cover the following properties and wondering if they have a formal name to search more efficiently. The function $f(x)$ cover the following ...
0
votes
1answer
33 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
32 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
52 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
43 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
21 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
26 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
15 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
22 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
40 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 ...