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

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1answer
86 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
3
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1answer
69 views

What is the domain of the following function?

Please tell me the domain of $y = \sin^{-1}(\sin(x))$ P.S. I think domain is $(-\infty, \infty)$ But my teacher says it is $(-\frac{\pi}{2}, \frac{\pi}{2})$. He says since $sine$ is a many one ...
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1answer
63 views

Is the source (and/or target) a group, or just its underlying set?

Consider the following statements. Let $G$ and $H$ denote groups and $f : G \rightarrow H$ denote an arbitrary function. Let $G$ and $H$ denote groups and $f : G \rightarrow H$ denote an arbitrary ...
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1answer
68 views

Fixed point and non-fixed point function

For constructing another proof I need two functions explicitly and therefore I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the ...
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2answers
549 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
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1answer
78 views

Relation $R$: $R\circ R \subseteq R \implies R$ is transitive

Let $R$ be a relation on $X$, a set. If $R\circ R\subseteq R$, then is $R$ transitive?
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7answers
421 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
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2answers
80 views

linearly independence of $e^{a_1x},… e^{a_nx}$

$a_1,\ldots,a_n$ are real different numbers. Prove that the functions $e^{a_1x},...,e^{a_nx}$ are linearly independent group in $Fun(R,R)$. My way to try to prove it: I assumed: $b_1e^{a_1x} + ...
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1answer
344 views

“range of function” vs “target of function”?

Page 14 of Fundamentals of Computer Graphics states that if we have a function like this: ...the set that comes before the arrow is called the domain of the function, and the set on the right-hand ...
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1answer
53 views

Numerical Analysis, build a contractive function

I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin. The goal of this exercise is to find a value alpha such that: ...
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0answers
56 views

Fast way to switch between lagrange and newtonian representation of polynomials?

I just wanted to know whether there is a fast algorithm to switch between these two representation methods of polynomials $\in \mathbb{R}[x]$? By Lagrange I am refering to: enter link description ...
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1answer
48 views

question about functions (asymptotic)

This is right? $f=\Omega(g)\Rightarrow2^f=\Omega(2^g)$? If not I'd like to get a Counter-example. Thank you!
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3answers
58 views

Reversed graph plotting algorithm

Consider a real-valued polynomial function of one variable. Knowing the "rules" function dictates we can plot the graph of that function with a given accuracy. Question: is the reversed process ...
2
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1answer
100 views

Name of the $(-1)^n$ function?

Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ? EDIT: The context of this question is that I need a name for this ...
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2answers
100 views

how to prove this equality

There are two equalities, $\sinh({\cosh }^{-1}x)=\sqrt { {x }^2-1 }\quad (x>1)$ $\cosh({\sinh}^{-1}y)=\sqrt{1+{y}^2}$ prove this equality please.. how to prove it? i cannot try it.. also, ...
4
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1answer
133 views

Get the number of digits from a number

I'm looking for a function $f$ that would give me the following results : For any $x$ such as $ x \in \mathbb {N^*}, x>0 $ $f(x) = 1 $ when $1\leq x < 10$ $f(x) = 2 $ when $10\leq x < ...
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0answers
116 views

Induction proof of surjectivity

I have a problem. Let $A: S\to T$ be a surjective map between finite sets. Prove by induction that $|S|\geq|T|$ and that if $|S|=|T|$, then $A$ is bijective. Another way to phrase the question is: ...
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1answer
500 views

Can anyone provide me a step-by-step proof for proving a function IS onto/surjective?

I've seen the definition, I've seen several examples and anti-examples (e.g. the typical x squared example). I get the idea, but I can't seem to find a proof for proving that a function IS onto, with ...
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2answers
87 views

Which of the following is not used to determine the slope of a function algebraically?

I dont know the answer to the above question. I think it is the slope of a secant line but I'm not sure ...
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3answers
819 views

Which function is appropriate for the geometrical shape mostly used as LOVE symbol

I tried $\;r=-1-\sin\theta,\,$ for $\,\theta$ between $[-2\pi,2\pi]$. But it's not as accurate as expected. Any one to help me? Thank you in advance.
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2answers
601 views

I need a better explanation of $(\epsilon,\delta)$-definition of limit

I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia. It says that $f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but ...
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1answer
34 views

Injectivity of a complex function

How come that $f(z)=z^3$ for $z\in \mathbb{C}$ is not injective on an open set $U$ with the origin deleted?
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1answer
40 views

How find the range of the $t_{1}-t_{2}$?

let $$f(x)=\begin{cases} \dfrac{1}{a-1}(x-1)&x\ge a\\ \dfrac{1}{a-2}(x-2)& x<a \end{cases}$$ There exist $t_{1},t_{2}$ such that $$f(t_{1})=\dfrac{1}{2},f(t_{2})=\dfrac{3}{2}$$ then ...
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0answers
100 views

What could be a monotonic, continuous and smooth function with these conditions?

I am looking for a smooth, continuous and monotonic function that satisfies these conditions: $f(0) = 0$, $f'(0) = 1$, $f(m) = 1$, $f'(m) = 1/m$, where $m > 1$ is a fixed real number. Also, it ...
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4answers
102 views

find minimum of given function

today my relative asked a problem,which had strange solution and i am curious, how this solution is get from such kind of equations. let say function has form $f(x)=a\sin(x)+b\cos(x)$ we should ...
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1answer
223 views

rotate graph of function by 180

suppose that we have graph of function $$f(x)=1+x\cos(x)$$ and we should rotate it by $180$ degree,question is what is a function which describe new graph?answer is $$f(x)=x\cos(x)-1$$but i can't ...
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1answer
77 views

ODE phase portrait and vector function interpretation

I do not quite remember how to plot a vector function (or maybe I do). Consider the ODE: \begin{equation} x' = \begin{pmatrix}1&1\\-1&1\end{pmatrix}x \end{equation} I have found the general ...
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2answers
149 views

find all rational functions $f(x)$ such that $(f(x))^2-f(x^2)=constant$,

Find all $f(x)$ such that $$(f(x))^2-f(x^2)=constant$$
4
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2answers
143 views

Use of Multiple “if and only if” statements

I apologize in advance if this question is too basic to warrant a post. I just ran into the following question: Let $f: A \to B$. If $A$ and $B$ are finite sets with the same number of elements, then ...
2
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0answers
187 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
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2answers
206 views

Proving The Average Value of a Function with Infinite Length

This is the given: One can extend the definition of the average value of a continuous function $f(x)$ to the interval $[a,\infty)$ of infinite length as follows: ...
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2answers
82 views

Trigonometry - Finding the range of the function

Problem : $$f(\theta)=(2\sqrt{3}+4)\sin\theta +4\cos \theta $$ I have studied if the function is in the form : $f(\theta)=a\cos\theta + b\sin\theta$ then the range of this function can be given as ...
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2answers
327 views

I need to find the difference quotient for a function

Here's the question: I am having difficulty finding the difference quotient for: $$y=f(x)=(x-2)^3+4$$
2
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1answer
146 views

Concave function properties

Given a concave function $f(x)$, $\,f(x)$ decreases as $\,x\,$ increases. That is, $\;f(x_1)\gt f(x_2)\,$ if $\,x_2\gt x_1$ For $\;f(x_1)+f(x_2)\;$ and ...
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1answer
148 views

What is the definition of the slope of a linear function in the context of economic graphs?

I only ask this because of the fact that economists tend to plot the dependent variable on the horizontal axis and the independent variable on the vertical, which is opposite to the "normal" way of ...
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1answer
44 views

A question on the domain of a complex function

Let $\log$ be the main branch. What is the domain of the complex function $f(z)=\log(1/(1-z^{2}))$?
5
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3answers
78 views

How do I transform $f(x)=\log(1+e^x)$ such that graph rotates $90^{\circ}$ on the $x$-$y$ axis

I am looking for a function $f(x)$ that is of a specific shape on the $x$-$y$ axis. I have a function $f(x)=\log(1+e^x)$ that has right shape. I want it rotated $90^\circ$ on $x$-$y$ axis. How can ...
4
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5answers
131 views

Is this true? $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$.

Is this true? Given $f,g\colon\mathbb R\to\mathbb R$. $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$. I met this problem when dealing with a coding method, but I'm really not ...
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3answers
104 views

How to construct a non-piecewise differentiable S curve with constant quick turning flat tails and a linear slope?

I need to find an example of a non-piecewise differentiable $f:\mathbb{R}\to\mathbb{R}$ such that $$ \begin{cases} f(x)=C_1 &\text{ for } x<X_1,\\ C_1 < f(x) < C_2 &\text{ ...
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1answer
222 views

Sum of terms in a composition cycle

Let $f, g$ be linear functions. Define $S(x)$ as $any$ composition sequence of $f$ and $g$ like $S(x) = (f\circ g\circ g\circ f\circ f\circ g)(x)$ Let $s$ as the fixed point of $S$ then a cycle is ...
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1answer
551 views

Finding the interval for increase of the function $y =x^2e^{-x}$

Problem : Find the interval in which the function $y =x^2e^{-x}$ is increasing . My approach : We can take first derivative to the find the increase or decrease of function ie. ...
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3answers
190 views

How do I find the domain of a function?

Given the function $h(x)=\sqrt {5-x} $ and the function $f(x)=3x^2+\frac {6 } {x } -8$ how would I find the domain without graphing?
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3answers
224 views

What does $f: A \times A \to A$ mean?

What does $f: A \times A \to A$ mean? Can you give some examples please?
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0answers
118 views

Upperbound of the minimum of the sum of two functions

Suppose you have two functions $f_1$ and $f_2$ and you know the minimum and the maximum values of each function. What's a good upperbound for the minimum value of $f_1+f_2$? I thought ...
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1answer
66 views

Help with graphing a piecewise function

What would be the graph and domain of this function? My domain is $(- \infty, \infty)$. I am stuck on graphing $-2x$. $$g(x)=\begin{cases} x+9 & \text{if }x<-3,\\ -2x & \text{if }|x|\leq ...
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1answer
78 views

domain of square root

What is the domain and range of square $\sqrt{3-t} - \sqrt{2+t}$? I consider the domain the two separate domain of each square root. My domain is $[-2,3]$. Is it right? Are there methods on how to ...
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1answer
81 views

Can you solve this problem with functions?

We are given $f(x)=ax^2+2x+b$, a is not $0$, $Df=R$ and $f\circ g=g\circ f$ where $g(x)=x$ has only solution $x_0$. Then we have to show that $ab\leq 1/4$.
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1answer
174 views

How do I create a sigmoid-esque function with the following properties?

For a range of $x$ values between $A$ and $B$ I would like $f(x) \rightarrow x$. For values less than $A$ I would like $f(x)$ to exhibit a sigmoid-esque convergence to $A'$ where $A'$ is $A - ...
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1answer
77 views

Uniform convergence of $\sin\left ( {\frac{1}{n^{3}x}} \right )$

I ran into this question and im not sure how to solve it: Check uniform convergence of: $$f_{n}(x)=\sin\left ( {\frac{1}{n^{3}x}} \right )\quad \;x\in (0,1]$$ I tried finding the supremum of the ...
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2answers
179 views

How to find the range of $f(x) = {e^x \over x-1}$

I want to find the range of the following function : $$f(x) = {e^x \over x-1}$$ How do I find the range of the above function ? I have tried a lot , but do not have any idea to solve this.