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

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20
votes
11answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
0
votes
1answer
78 views

Prove expectation finite

Let $ \{b_n\} $ be a sequence of non-zero complex numbers. We have $ N(t)=\#\{n \geq 1:|b_{n}|\leq t\} $, $ \displaystyle\limsup_{t\rightarrow\infty} N(t)/t^p<\infty (1\leq p <2)$, for $ X_1\in ...
2
votes
2answers
34 views

Range of an inverse trigonometric function

Find the range of $f(x)=\arccos\sqrt {x^2+3x+1}+\arccos\sqrt {x^2+3x}$ My attempt is:I first found domain, $x^2+3x\geq0$ $x\leq-3$ or $x\geq0$...........(1) $x^2+3x+1\geq0$ ...
1
vote
1answer
77 views

Is it possible to define $x+x+x+x…x$ times? [duplicate]

Is it possible to define $x+x+x+x...x $ times? I need to compute its derivative. It differs from the derivative of $x^2$. It evaluates to $x$ via sum of derivatives.
2
votes
1answer
40 views

Function of sin x

Give that $f(x)=\sin x$ for the domain $0\leq x \leq k$, find the greatest value of $k$ for which $f(x)$ has an inverse. Is the answer $\frac{\pi}{2}$?
0
votes
1answer
16 views

How can you test if a function is bounded by another?

I am trying to learn about complexity theory, which states that $f(n)$ is in $O(g)$ if, for some $C > 0$, $f(n) \leq C\cdot g(n)$ for all $n \in \mathbb{N}$. That's well and good; it makes sense to ...
0
votes
2answers
23 views

Find k for which the equation has equal roots.

I find to find the value of k in terms of $\alpha$ and $\beta$ . I rearranged the equation $x^2 +kx-1=2k+x$ into $x^2+(k-1)x-(1+2k)=0.$ I then found $\alpha$$\beta$ to equal$-1-2k$. and $\alpha + ...
0
votes
1answer
14 views

Naka-rushton function

I am trying to figure out how to transform the naka-rushton equation to a S-shaped function. The naka-rushton equations is defined by \begin{equation} R(C) = R_{max}\frac{C^n}{C^n+K^n}+b, ...
0
votes
2answers
37 views

inverse a function with exponential and first degree polynom

I need some help to inverse this function: $$ y = a(e^{bx}-1) + cx + d $$ with $y(0)=d$ and $y(k)=0$ where $k$ is a constant. I don't know how to proceed. Thanks.
-1
votes
2answers
25 views

Properties of a certain binary relation [closed]

$R$ is a binary relation function $(x,y) \in \mathbb R^2$. If $$ R = \{(x, y)\in\mathbb R^2\mid \lfloor x\rfloor = \lceil y\rceil\} $$ then: Is $R$ reflexive, irreflexive or neither Is $R$ ...
2
votes
1answer
58 views

How to find the Summation S

Given function $f(x)=\frac{9x}{9x+3}$. Find S: $$ S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right) $$
0
votes
0answers
57 views

A weird problem on expected value of a random variable [closed]

Let $X$ be a discrete random variable taking values $x_1, x_2$, ... with probabilities $p_1, p_2$, ... respectively. Then the expected value of this random variable is $E(X)=\sum_{i=1}^{\infty }x_i ...
0
votes
1answer
13 views

Definition of sigmoidal curve with epsilon

I want to create a sigmoidal curve $f(x)$ with the parameters $s$ and $\epsilon$ so that it has the following features: $f(0) = 0 +\epsilon$ $f(s) = 1 - \epsilon$ $f'(s/2)=1$ Is this possible? If ...
-1
votes
1answer
56 views

How to find the inverse of $f(x)=2x - x^2$? [closed]

What is the inverse of $f(x)=2x - x^2$ in the domain (0,1) and the range (0,1)?
2
votes
3answers
55 views

What is the value of this function?

Consider the three-variable function defined at the following way for all natural numbers $n,x,y$ : $f(0,x,y) = x+y $ $f(n,x,0) = x$ $f(n,x,y) = f(n-1, $ $ $ $f(n,x,y-1) , $ $ $ $f(n,x,y-1)+y ) $. ...
0
votes
1answer
49 views

Functions with real domain but complex range, do they have any use?

For example if we define the square root function like this: $$\text{Sqrt}({x})= \begin{cases} \sqrt{x} & x\geq 0 \\ i\sqrt{-x} & x<0 \end{cases}$$ Or we could have an exponential ...
0
votes
1answer
19 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
3
votes
2answers
78 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
1
vote
1answer
50 views

Can you help me with a function with cos and radians?

I have this function: $$d(n)=15-2,5\cdot\cos\left(\pi\frac{n-31}{360}\right).$$ This function talks about the difference between when day start and day over. When I change $n$ for $5$ result for me ...
0
votes
0answers
14 views

Converting Properties to Specific Function

Is there an area in mathematics that deals with the idea of deriving functions from a given set of certain desired properties? The concept seems very important. You can read about the history of ...
1
vote
1answer
40 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
2
votes
2answers
45 views

If $ f(x) $is convex then $ yf(x/y)$ is convex

I am struggling with this question: Show that if $f(x)$ is convex then the function $ yf(x/y)$ is convex on $\{(x, y): y>0\}$. I have tried starting from the standard definition of convexity but ...
-3
votes
3answers
30 views

Show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R and show that $x^2-x+1$ has same sign for all values of x [closed]

How to show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R How to show that $x^2-x+1$ has the same sign for all values of x.
-2
votes
0answers
58 views

Does $f(x)=x^3+x^4\sin\frac{\pi}{x}$ have the inflection point? [closed]

Does the function $f(x)= x^3+x^4\sin\frac{\pi}{x} \,\,\,$ have an inflection point? Is it $(0,0)$?
2
votes
4answers
380 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
3
votes
1answer
63 views

Real Analysis: Given $A_1$, $A_2$, $A_3$, which set is open, which is closed?

a. $A_1 = \{x: \sin(1/x) =0\}$ Clearly $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, what can we say about this set? b. $A_2 = \{x: x\sin(1/x)= 0 \}$ Again $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, is ...
-3
votes
1answer
16 views

Sketching the graph of a sequence of functions [closed]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
1
vote
0answers
8 views

baire $1$ function

Here is a new definition of Baire Class one function. Suppose that $X$ is a complete separable metric space. A function $f:X \rightarrow \mathbb{R}$ is said to be Baire class one if for any ...
0
votes
1answer
46 views

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ for all $x,y\in\mathbb{Q}^+$. Before this problem, I have solved few similar ...
6
votes
0answers
80 views

Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
-3
votes
0answers
44 views

Solving trigonometirc functions [closed]

I am having difficulty understanding this question. I have tried to find the solution but am coming up with a negative answer and the answer must be positive as t in this equation is time. I am not ...
0
votes
0answers
16 views

Bijective Function from n-fold Cartesian Product to set of functions

This is my first posting, so I apologize for the cumbersome formatting; I am not yet familiar with MathJax/Latex commands. Let $Y$ be a set and let $n$ be an element of $\mathbb N$ (natural numbers). ...
0
votes
1answer
30 views

How to solve for k to get equal roots in a composite function

I am given: $f(x) = 4x-2k\ $ and $\ g(x) = \dfrac{9}{2-x}$ ($x \not = 2$) The question is: Find the values for k for which the equation $fg(x) = x$ has two equal roots. So $f(g(x))=\dfrac{4\cdot ...
1
vote
2answers
47 views

Proving two functions are monotonically related

I've been relying on stackexchange a lot lately but this is my first time asking a question. A lot of searching has yielded no answer so hopefully someone can help out. I'm trying to find (and prove) ...
0
votes
1answer
54 views

Find the inverse function of $ f (x ) = x^2 - x - 2$

Find the inverse function of $ f (x ) = x^2 - x - 2$, where x is equal to or larger than 1/2. I tried to express it in form of $ (x - 1 )^2 = y + 2 $, but this is not true as the term in the middle ...
3
votes
1answer
33 views

biggest possible domain of diffeomorphism

Consider the function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2, (x,y) \mapsto (x^2-y^2,2xy).$$ How can I determine a subset $D$ of $\mathbb{R}^2$ as big as possible such that $f|_D$ becomes a ...
0
votes
1answer
24 views

Don't understand this question [table of ordered pairs, find missing values]

I am very confused by this question that I have encountered while practicing for my GED. Over the last 6 months or so I've taken 3 official practice tests, and every time I took a test I encountered ...
4
votes
1answer
34 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
0
votes
1answer
14 views

Creating Polynomial Function with Surface Area of Cylinder

I've spent a few hours at this question but can't seem to get the right answer. I was hoping someone here can lead me in the right direction. The question: A storage tank is to be constructed ...
0
votes
1answer
30 views

Does the existence of a maximum over $|f_u(x,u(x))|$ imply that $f$ satisfies the Lipschitz condition?

A function $f$ satisfies the Lipschitz condition if: $$|f(x,u_1(x)) - f(x,u_2(x))| \leq A |u_1(x) - u_2(x)|$$ $$\frac {|f(x,u_1(x)) - f(x,u_2(x))|}{|u_1(x) - u_2(x)|} \leq A $$ Does the existence ...
-2
votes
0answers
57 views

What is this function? [closed]

I've seen some formula that seems to make start value approach to final value. Probably the formula is incorrect, but I believe that here, people will recognize what is that and write correct. Please, ...
2
votes
1answer
38 views

Substitution with functional equations

I've found this nice introduction worksheet that I started to work through with the goal to get a better understanding of functions and finding them in equations. I've gotten so far but in this one ...
0
votes
1answer
28 views

How to derive the general formula to determine the equation of a given cubic function

My question is: When determining the equation of a cubic function, we can separate the general cubic equation into it's solutions and we end up with the equation $y = a(x-r_1)(x-r_2)(x-r_3)$ We ...
0
votes
2answers
18 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
0
votes
0answers
24 views

Definition of positive definite function

I'm trying to understand a definition of a positive definite function I found in a book on Lyapunov stability Definition: A continuous function $W(x)$ is said to be a positive definite function if ...
0
votes
0answers
20 views

Derivative of a correlation function

From a big set of data I create a correlation function between a response parameter and three input parameters $(P_1, P_2, P_3)$. $Response = K_1 + K_2 \cdot P_1 + K_3 \cdot P_2 + K_4 \cdot P_3 + ...
0
votes
0answers
33 views

difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized ...
1
vote
1answer
37 views

Discrete mathematics, equivalence relations, functions.

I'd like some insight on how to 'solve' this problem (more towards understanding what the problem is asking) Suppose that $A$ is a nonempty set and $\mathcal{R}$ is an equivalence relation on ...
1
vote
1answer
33 views

Function Domain of cubic root [duplicate]

Does the domain of the function $y=\sqrt[3]{x^3+1}$ include $x<-1$? If yes, why is Mathematica and Wolfram Alpha not plotting that part of the function?
-4
votes
3answers
33 views

a question of mapping and set theory [closed]

A set $S$ is said to be infinite if there is a one-one correspondence between $S$ and any proper subset of $S$ prove The set of integers is infinite . The set of real number is infinite. If a set ...