Tagged Questions

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

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4
votes
4answers
257 views

Functions of algebra that deal with real number

If the function $f$ satisfies the equation $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, what are the possible values of $f(0)$? A.  Any real number B.  Any ...
0
votes
4answers
50 views

Algebra that includes functions and graphing

The answer to the following is B. Can someone explain me how it is please?
0
votes
0answers
61 views

The parent function of this particular graph

I'm looking for the rough "parent function" of the following graph. The graph has a vertical asymptote and a horizontal asymptote. It is most similar to f(x) = ln(x), but more "compressed." Anyone ...
1
vote
1answer
174 views

Is this function convex or concave on $(x,y,z)$?

Is this function convex or concave on $(x,y,z)$? $A$, $B$, $a$, $b$, and $c$ are positive constants. $$f(x,y,z) = A\exp\left(\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}\right) + ...
0
votes
0answers
157 views

From point-wise to essential supremum of a set of real-valued measurable functions

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$. The problem is, that the sets involved are uncountable and thus, the point-wise ...
4
votes
0answers
158 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
0
votes
3answers
91 views

How to show that $\frac{z-a}{a-z}$ has not inverse?

How to show that $\frac{z-a}{a-z}$ has not inverse? I know that $$\frac{z-a}{a-z}=\frac{-1(a-z)}{a-z}=-1 .$$ But state if I'm wrong, that following is true: $$f(z)=\frac{z-a}{a-z} \Leftrightarrow ...
0
votes
2answers
240 views

Addition function injective?

I am just curious to know if addition of two numbers an injective function? Lets say $\operatorname{Sum}(a,b) = a + b$ Now is the $\operatorname{Sum}$ function an injective functions?
1
vote
1answer
40 views

The shape of the functions

How many of the following functions on R are increasing on their domain? $y = e^x$, $y = x^2$, $y = x^3$ (a) 0 (b) 1 (c) 3 (d) 2 How many of the following functions on R are concave up on their ...
-1
votes
1answer
67 views

Functions Questions…

Hey im doing some functions questions and im looking at my notes and i cant figure out how to do them. I also have answered the first 3 and i would appreciate if anyone could tell me if i am ...
0
votes
3answers
90 views

Construct function with 2 local minima at x1 and x2

I am trying to construct a continuous differentiable function $f(x)$ that for $x_1$ and $x_2$ takes the value $0$ and have global minimum at these points, i.e. $f(x_1)=f(x_2)=0$ and ...
1
vote
2answers
245 views

Create asymmetric 3D function by revolving a 2D function around an axis

Could someone please let me know how I can construct the equation for asymmetric torus similar to the figure below? The asymmetric torus seems to be a 2D function revolved around an axis while being ...
3
votes
1answer
136 views

How to bound the maximal consecutive length in a random subset of [n] as function of n?

Let $S$ be a random subset of $[n]=\{1,2,\ldots,n\}$ chosen uniformly from $[n]$'s subsets. How can I find a function $f(n)$ s.t. for any $\varepsilon \gt 0$, $$\lim_{n \rightarrow \infty} P\left[(1- ...
5
votes
4answers
322 views

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded? ($a$ and $b$ being finite numbers). I tried proving and disproving it. Couldn't find an example for a ...
1
vote
1answer
126 views

inverse of function

Thanks for the help! Here is the solution.. i had a problem: $$f(x)=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$ i had to find the inverse, so lets begin... 1) first i write in terms of $y$ ...
0
votes
3answers
194 views

The square of a measurable function is measurable

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a measurable function. I want to show that $f^2:x\mapsto (f(x))^2$ is measurable. Apparently it can be shown using the facts that the sum of two ...
0
votes
2answers
98 views

Notation for probability of either sign

I have a function $f(k) = \pm 2^k$ with probability $1/2$ of either sign. How would I express this in a cleaner notation? I'm guessing to use the Kronecker delta somehow, but I can't put a finger on ...
4
votes
3answers
196 views

name this function

Is there a function that has these properties? Points: $f(1)=\tfrac{1}{2}$ $f(-1)=-\tfrac{1}{2}$ $f(0)=0$ Bounds: $f$ is bounded between $(-1,1)$: $\forall x\in\mathbb{R}: -1 < f(x) < 1$ ...
2
votes
2answers
188 views

Math symbol for approximation of probability distribution by arbitrary function?

I want to use a symbol between two functions; $$p\text{ (symbol) }f$$ such that $p$ is a probability function and $\text{(symbol)}$ implies: we do not have access to $p$ but we approximate it with ...
1
vote
1answer
41 views

How do I approach this signals and systems question?

The question seems very abstract to me as it doesn't really describe what it is asking for. The problem states the following, let: $$x[n]=\begin{cases} n,&\text{if n is odd};\\\\ ...
1
vote
3answers
163 views

Reference request - being rigorous about a common abuse of notation.

I've completely rewritten this question, in accordance with this advice. As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
0
votes
1answer
2k views

What does it mean when you say that the function is bounded?

What I figured is that it means that the function has an upper bound, however I came across this text: Here since g(x) either equal or less to f(x), |g(x) / f(x)| must be bounded right? Since the ...
2
votes
1answer
210 views

Convex function inequalities

1) Let $f:\mathbb{R}\to\mathbb{R} $ be a positive, convex, continuous function. Assume $f$ satisifes the following inequality$$f(x)f(y)\leq f(xy)$$ for all $x,y\in \mathbb{R}.$ What can we say most ...
0
votes
1answer
82 views

What is the most effective way to implement Hilbert's hotel? [duplicate]

Assuming I need to find an onto and 1-to-1 function from $(a,b)$ to $(0,1)$, well that's not a hard job. But things are getting bit more complicated when I'm asked to do the exact same but from ...
1
vote
2answers
1k views

Formula for Snake Draft pick numbers

Hello I am trying to come up with a formula to calculate the overall pick number in a snake style draft. For example in a snake draft every other round the pick order reverses. So in a 10 team league ...
1
vote
1answer
37 views

A question of formality regarding limits

Let $f$ be a differentiable function over $\mathbb{R}$, with its derivative being a continuous function. Let there be a function $g$ s.t. $\lim_{x \to 0} g(x) = 0$. Now I'm required to show that ...
1
vote
1answer
130 views

What's the difference between T(V) and ImT?

Assuming I have the following Linear transformation: $\mathbb{T}: \mathbb{V} \rightarrow \mathbb{W}$ where $\mathbb {V}$ and $\mathbb {W}$ are vector space. ...
4
votes
2answers
681 views

how can we convert sin function into continued fraction?

how can we convert sin function into continued fraction ? for example http://mathworld.wolfram.com/EulersContinuedFraction.html how can we convert sin to simmilar continued fraction ?? and what ...
4
votes
1answer
313 views

Find a translation that fixes the graph of cosine function

Find a translation that fixes $y=\cos x$ That is, the goal is a translation of the plane that fixes this curve: I have tried this numerous times but can't seem to find a method of doing it. ...
2
votes
3answers
553 views

Proof that $(1+1/x)^x$ is monotonic increasing

How does one prove that $(1+\frac{1}{x})^x$ is monotonic increasing for any $x \in [1,\infty)$? Thanks a million!
3
votes
1answer
348 views

How is the following function an odd function? $S(x) = \sin x/x$, $x \neq 0$

How is the following function an odd function? $S(x) = \frac{\sin x}{x}$, $x \neq 0$ I get $$\frac{\sin(-x)}{-x} = \frac{\sin x}{x}$$ which is even right? because $S(-x) = S(x)$? So unless the ...
1
vote
1answer
1k views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
1
vote
0answers
85 views

Function on equivalence relation on functions

Let $E$ be the set of all functions from a set $X$ into a set $Y$. Let $b \in X$ and let $R$ be the subset of $E \times E$ consisting of those pairs $(f,g)$ such that $f(b) = g(b)$. Prove that $R$ is ...
2
votes
1answer
191 views

Equivalence relation function

Let $f:X \to X$ be an injective function from a set $X$ into itself. Define a sequence of functions $f^0 , f^1, f^2, \dots : X \to X$ by letting $f^0 = \mathrm{id}$, $f^1 = f$ and $f^n = ...
2
votes
4answers
90 views

How to show that $ n^{2} = 4^{{\log_{2}}(n)} $?

I ran across this simple identity yesterday, but can’t seem to find a way to get from one side to the other: $$ n^{2} = 4^{{\log_{2}}(n)}. $$ Wolfram Alpha tells me that it is true, but other than ...
2
votes
1answer
487 views

Injection function proof

Suppose $f$ is an injection. Show that $f^{-1}\circ f(x)=x$ for all $x\in D(f)$ and $f\circ f^{-1}(y)=y$ for all $y$ in $R(f)$. In $f^{-1}$ it is defined as "Let $f$ be a one-one function with ...
0
votes
1answer
43 views

Elementary function and injection

I am a bit confused on the definition of a function and an injection. I can't seem to distinguish the two. The definition of a function is: if $(x_1,y_1)\in f, (x_1,y_2)\in f$ then $y_1 = y_2$, ...
2
votes
5answers
583 views

Can I really factor a constant into the $\min$ function?

Say I have $\min(5x_1,x_2)$ and I multiply the whole function by $10$, i.e. $10\min(5x_1,x_2)$. Does that simplify to $\min(50x_1,10x_1)$? In one of my classes I think my professor did this but I'm ...
0
votes
3answers
338 views

Finding a monotonically increasing function with limit 1

To polish/improve a homework answer, I am trying to find a monotonically, continuous, strictly increasing function $f$ with these properties: $f(0) = 0$ $\lim_{x \to \infty} f(x) = 1$ (I don't ...
2
votes
1answer
131 views

Reconstructing a Paragraph From Random Set of Words

Goal: Take a collection of randomly shuffled words that represent x and y coordinates on a plane, re-order them such to construct the original paragraph those words came from. Each word represents ...
-1
votes
5answers
108 views

An inverse of the function $e^x$

How can I prove that the function $L(x)=\int_1^xdt/t$ which is definte on $(0,\infty)$ is an invers of $\exp(x)$. Should I work on $L(x)\circ e^x=e^x\circ L(x)=x$. I am stuck.... Thanks.
0
votes
1answer
53 views

Simple bijection: help please

I am trying to show that if two sets $A,B$ have $n$ and $m$ distinct elements respectively then $A \times B$ has $nm$ elements. I assumed that there are bijections $f:\{1, ...,n\} \to A, k \mapsto ...
1
vote
2answers
153 views

Stretching a curve towards one general direction without changing two points in a curve

Hi, I'm trying to "stretch" the following curve $f(x) = \ln(x + 1)$ so that it appears similar to the green curve I have drawn in the above picture. The blue arrow indicates which general direction ...
0
votes
1answer
147 views

Computing functions from generating functions

I am new to generating functions but understand how to derive them from given discrete numeric functions. Is there a simple way to derive the discrete numeric function given a generating function. For ...
2
votes
1answer
97 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
0
votes
1answer
145 views

A discrete function and its rate of oscillation

Consider a function $y[n]= \cos[w n ]$, where $n$ is an integer. I have to prove that this signal will have highest rate of oscillation at $w = \pi$. I was thinking I can take the derivatives ...
2
votes
1answer
547 views

How can I determine the similarity of these graphs/curves?

I have 3 visually similar graphs pictured below. They have similar peak patterns that are visible to the naked eye, but I want to compare their similarity mathematically. I can sum each column to ...
0
votes
2answers
178 views

Help me to find function to this graph

I've got this function: $\frac{x^2}{x^2+(1-x)^2}$ ; it gives me this blue graph (in zero - one range): Could you help me find function to achieve graph close to red one?
0
votes
2answers
73 views

Continuous function, not sure what to do here…

The question is as follows: Let $f(x) = \begin{cases} x, & \mbox{if } x<1 \\ x^2+1, & \mbox{if } x\ge 1 \end{cases}$ Let $g$ be a function such that $fg$ is continuous at $1$, and ...
1
vote
1answer
29 views

Is the function is differentiable at $x$ or $D$?

I know that a) and b) is differentiable at the given points, would you maybe explain how I should show that ? a) $f:\mathbb{R}\rightarrow \mathbb{R},\quad x\rightarrow 0,{ \quad x }_{ 0 }=0$ b) ...