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

learn more… | top users | synonyms (1)

0
votes
2answers
143 views

Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
1
vote
1answer
49 views

Elevation of 3D function

$f(x,y) = \begin{cases} x^2/y & y \neq 0 \\ 0 & y = 0\end{cases}$ I need to draw the elevation (or you may call it Equivalent curve) of this function and I don't know how to draw them. Can ...
0
votes
2answers
152 views

The number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$ (Putnam 1993)

Let ${\cal P}_n$ be the set of subsets of $\{1, 2, \dots, n\}$. Let $c(n, m)$ be the number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$. Prove that ...
0
votes
1answer
47 views

Functions operating in uncountable sets with cardinality $\gt\aleph_1$

A generic function $y=f(x)$ maps a number in the set of real number $X$ in another number in the set $Y$. It's well known that the irrational numbers are not countable. It's also known we can get a ...
0
votes
2answers
158 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
25
votes
16answers
3k views

Produce unique number given two integers

Given two integers, $a$ and $b$, I need an operation to produce a third number $c$. This number does not have to be an integer. The restrictions are as follows: $c$ must be unique for the inputs ...
4
votes
1answer
70 views

Existence of a function with boundary conditions for derivatives

Does there exist a function $f\in C^2(\Bbb{R},\Bbb{R})$ such that $\frac{f'(x)}{f(x)}\rightarrow_{x\rightarrow\infty}+\infty$ but $\frac{f''(x)}{f'(x)}\rightarrow_{x\rightarrow\infty} 0$ ? I know for ...
2
votes
2answers
90 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
1
vote
1answer
51 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
1
vote
1answer
38 views

Function on plane with incenter

Let $f$ be a function from the set of all points on the plane to the nonzero real numbers. Suppose that for any triangle $ABC$ with incenter $I$, we have that $f(I)=f(A)f(B)f(C)$. What are the ...
1
vote
1answer
67 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
0
votes
1answer
33 views

How to prove subsets of a function

$ f: \mathbb R \to \mathbb R$ is defined as a one to one function. For any collection of subsets $A_1, A_2, A_3 ......A_n$ prove that, $$ f(A_1 \cap A_2 \cap A_3 ......A_n) = f(A_1) \cap f(A_2) \cap ...
1
vote
1answer
106 views

Proving f(rx) = rf(x)

What's the difference between the proofs of $$ f(rx)= rf(x) \forall r \in \mathbb Z ,\forall x \in \mathbb R $$ and $$ f(rx)= rf(x) \forall r \in \mathbb Q , \forall x \in \mathbb R $$ where $ f : ...
0
votes
2answers
56 views

Matrix of Functions

I was thinking about a problem and I realized for a family of functions say $L^1([a,b])$ one can define matrices with functions as elements: $$ A=\left[ \begin{array}{ccc} f_{11}& ...
2
votes
2answers
72 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
0
votes
1answer
29 views

Even function about a point over a restricted range

Why is $f(x)=(x-1)^2$sin$(n\pi x)$ even about $x=1$ for $0\leq x \leq2$? I understand that $(x-1)^2$ is even about $x=1$ and I can plot the graph for various values of $n$ on wolfram alpha, but how ...
1
vote
2answers
80 views

Setup Quadratic Word Problem

I need help setting up this quadratic word problem, I have no idea where to start. Among all pairs of (real) numbers whose sum is 17, find a pair whose product is as large as possible. What is the ...
0
votes
1answer
51 views

Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
3
votes
1answer
298 views

Exact value of expression

Let $$f(x)=\frac{4^x}{4^x+2}$$ and $$S=\sum_{n=1}^{2005}f\left(\frac{n}{2005}\right)$$ What is the exact value of $S$? I tried to write $a=4^{\large\frac{1}{2005}}$, then ...
1
vote
3answers
39 views

How many solutions to quadratic logarithms?

For a given Log equation with a quadratic $x$, such as $$F(x)=\log(x^2)$$there appear to be two $x$ values, for every $F(x)$, a positve and a negative. However, if $F(x)$ is ...
1
vote
0answers
21 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
2
votes
1answer
334 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
vote
1answer
26 views

Function that has potential to increase current value, based on current value

Math is not my strong suit, let's start there. (Be gentle.) I have a game engine that is "ticking" every 1 second. I would like for a number, A, to increase at an ...
4
votes
2answers
110 views

Why is the period of $f$, $\pi$?

I came across a problem, which asked to compute the period of the function $$f(x)=3^{\sec^2x-\tan^2 x}.$$ The answer provided was $\pi$. I don't get how.
0
votes
4answers
79 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
1
vote
1answer
42 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
2
votes
1answer
125 views

If a function is both upper and lower semicontinuous, does it have to be continuous?

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] ...
0
votes
1answer
51 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
0
votes
1answer
52 views

Finding a population Function

I have been given the population of the USA from 1790 - 1980 (increasing in intervals of $10$) and I am asked to solve this differential equation. Using $t$ as time in years, and $P$ as size of ...
0
votes
2answers
80 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
0
votes
0answers
177 views

Function's graph sketch

$f(x) = x+9$ $g(x) = 2x-3$ I need to draw the sketch of $\min(f,g)$ and $\max(f,g)$. I tried: http://sketchtoy.com/62375984 The yellow is the min and the blue is max. The lines should be ...
0
votes
2answers
44 views

Getting a diverse set of three numbers from two numbers

I'm using this information to build an interface to pick a color, but I feel that this question is purely math-related. Please correct me if this is the wrong StackExchange site for this. I am making ...
0
votes
2answers
68 views

Geometric meaning of results obtained in (a) and (b)

The task: Plot the function $\sqrt{1-x^2}$. What does it look like? What is the geometric meaning of the results you obtained in (a) and (b)? Can anybody help me with geometric mean? I can't ...
1
vote
2answers
65 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
0
votes
1answer
52 views

How to make clear a letter is a function?

How should I make clear that a symbol is a function? Usually a function is denoted by the letter $f$ or $g$, or is directly applied to arguments (e.g. $c(x,y)$) or is implied to be a function by an ...
0
votes
2answers
63 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
vote
2answers
47 views

Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this. I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one ...
6
votes
2answers
98 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
-1
votes
3answers
38 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
0
votes
1answer
22 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
0
votes
1answer
43 views

Figuring the function $f(x)$ from given information

Here is the given information in my question, So, what my question inform is that there is a cubic polynomial function (i.e $f(x)$) which has local maxima at $x=-1$. While that for $f'(x)$, it's ...
0
votes
0answers
44 views

Iterative function eventually reaching identity

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that ...
0
votes
2answers
44 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
0
votes
1answer
50 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
1
vote
1answer
56 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
1
vote
0answers
31 views

Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
5
votes
1answer
87 views

Proof that there is a bijection, if there are injective maps in both directions

Let $A$ and $B$ be two sets. Let $f:A\to B$ be injective such that $Im(f) \subsetneq B$. Let $g:B\to A$ be injective such that $Im(g) \subsetneq A$. Obviously $A$ and $B$ are not finite sets. Can ...
2
votes
1answer
90 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
2
votes
4answers
76 views

Heaviside Unit Step Function

Convert to heaviside function: $$f(t) = \begin{cases}e^t ,& 0 \leq t \leq 1 \\0 ,& t > 1\end{cases}$$ My attempt: $f(t) = U(t) e^t - U(t-1) e^t $ I think my solution is not right because ...
8
votes
1answer
676 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.