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

learn more… | top users | synonyms (1)

1
vote
4answers
93 views

Finding injective mapping $f : \{0,1,2\}^\mathbb{N} \rightarrow \{0,1\}^\mathbb{N}$

I know that $g:\mathbb N\rightarrow\{0,1,2\}$ is a sequence. For example $a=(a_0,a_1,a_2,...)$ where every $a_i$ belongs to $\{0,1,2\}$ and I have to change it in to another sequence $b \in ...
0
votes
1answer
36 views

Why is this function within the $\Bbb{R^2}$ domain?

From curiosity I've seen the physical existance of the function: $$z=x·y$$ After going to Wolfram Alpha site I've seen that function happens to belong to the $\Bbb{R^2}$ domain. However, including ...
1
vote
2answers
54 views

How to calculate $E[X^2Y^5]$ given density functions for $x$ and $y$

Let $X$ and $Y$ be random independent variables within the limits $[0, 1]$ with the following density functions: $f_X(x) = 0.16x + 0.92$ such that $x$ is within the parameters $[0, 1]$ and $f_Y(y) = ...
0
votes
1answer
55 views

group homomorphism?

Given: $ \varphi : ( S_4, \circ) ~~~\rightarrow ~~~(\Bbb{Z}/4\Bbb{Z}),$ $~~~~~~~~~$ $\sigma \longmapsto [\sigma(1)]$ Is this function a group homomorphism? My idea is, to calculate some values, but ...
0
votes
1answer
139 views

If derivative of a function is non-zero then it is monotone. Since function is monotone, variable can be substituted in integration

I came across this in the text Differential Equations-An Introduction With Applications, by Lothar Collatz: $y'(x)=\frac{dy}{dx}=\frac{f(x)}{g(y)}$ Suppose $f(x)$ is continuous in $[a,b]$ and $g(y)$ ...
1
vote
1answer
95 views

Deriving time and distance

The distance an aircraft travels along a runway before takeoff is given by $D=(10/9)t^2$, where $D$ is measured in meters form the starting point and $t$ is measured in seconds from the time the ...
0
votes
1answer
178 views

Proving that a function is a contraction

The question is: Find values of $a$ such that the function $f(x)=ax^2 -1$ is a contraction on the interval $[1,2]$. I looked up the definition of a function being a contraction on the interval and ...
0
votes
1answer
28 views

Parametric Curves Existence of Tangent

If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$? Indeed, this is a very simple question. Sorry but I'm just a beginner for ...
0
votes
2answers
40 views

convexity of a function and inequality condition

i have two functions, a quadratic: $f(x) = ax^2 + bx + c$ (where $a>0$, $b<0$, $c>0$), and a linear function: $g(x) = dx + c$ (where $d<0$) now $f(0)=g(0)=c$ how can i use convexity of ...
0
votes
2answers
97 views

If $B \subseteq A$ and $f:A \to B$ is 1-1, it must be onto

Let $B \subseteq A$ and $f: A \to B$ be a 1-1 function, then $f$ must be onto. I understand that $f$ is onto if and only if every element of $B$ is in the image of $f$... I believe this ...
1
vote
1answer
76 views

Prove that f*g is differentiable at x0.

Let f,g : R-->R. Let f(x0) = 0, f(x) differentiable at x0 and g(x) continuous at x0. I need to prove that f*g is differentiable at x0. Any ideas of hints about how to begin? *continuousity doesnt ...
3
votes
1answer
13k views

Finding points on graph with tangent lines perpendicular to a line

Find all points $(x,y)$ on the graph of $y=\frac{x}{x-3}$ with tangent lines perpendicular to the line $y=3x-1.$ My thoughts on this problem: First I should find the slope of the given line ...
1
vote
1answer
96 views

prove that: $\lim_{x \to \infty} [f(x+1)-f(x)] = 0$ just by using definitions of limit and definition of derivative.

Let f and it is given that $\lim_{x\to \infty} f '(x) = 0$. I have to prove that: $\lim_{x\to \infty} [f(x+1)-f(x)] = 0 $ just by using definitions of limit and definition of derivative. I have no ...
1
vote
1answer
30 views
4
votes
2answers
88 views

Prove that $\lim_{n\to\infty}|f(x_n+\xi)-f(x_n)|=0$

Let $f:(0,+\infty)\to\mathbb{R}$ be continuous and bounded. Let $\xi>0$. Show that there is a sequence $(x_n)$ in $(0,+\infty)$ with $x_n\to\infty$ s.t. $$\lim_{n\to\infty}|f(x_n+\xi)-f(x_n)|=0.$$ ...
2
votes
3answers
985 views

Floor and ceiling function proof

I have the following to prove: $$\lfloor 3x\rfloor = \lfloor x\rfloor + \left\lfloor x+\frac 13 \right\rfloor + \left\lfloor x+\frac 23 \right\rfloor $$ The definition of a floor function is: $ ...
2
votes
1answer
77 views

Mathematics sign for ceil and floor functions

What is mathematics sign for ceil and floor ?
0
votes
1answer
27 views

Validity of notation from the aspect of function description

I have the following notation that should describe the nature of my function $for \forall a \in A \exists f:A \rightarrow S, A \subset N, S \subset [0,1]^n,|S|=n$ Can anyone tell me is the notation ...
1
vote
2answers
51 views

Solving a bijective function task

How would you solve this task actually? This is not homework btw, i'm just trying to understand how you can solve tasks like this. Let $S$ be the set $\{$$1,2,3,4,5$$\}$. How many functions from ...
0
votes
1answer
60 views

if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?

If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$ Then $f(0)= ?$
3
votes
1answer
38 views

Show there are distinct $\xi,\eta$ s.t. $\frac{a}{f'(\xi)}+\frac{b}{f'(\eta)}=a+b$

Let $f:[0,1]\to\mathbb{R}$ be continuous. Suppose $f$ is differentiable on $(0,1)$ and $f(0)=0,f(1)=1$. Show that for any positive real numbers $a,b$, there are distinct points $\xi,\eta\in(0,1)$ s.t. ...
1
vote
1answer
56 views

Confusion with this definition of the derivative

This function is from my text: $$p(\theta) = \sqrt{13\theta}$$ It states that the derivative of the function $p(\theta)$ with respect to the variable $\theta$ is the function $p'$ whose value at ...
1
vote
0answers
31 views

Coefficients of rational involution.

Question: (Spivak Calulus 3rd, Chapter 3, Problem 8) For which numbers $a,b,c,d$ will the function $f(x) = \frac{ax + d}{cx + b}$ satisfy $f(f(x)) = x$ for all $x$? Attempt at an answer: I tried ...
0
votes
1answer
63 views

Define Matlab function depending from another function

I have a function $\phi(x,t)$ and I want to define a function $f$ whose values depend on $\phi$. Specifically: $$f(x,t) = \begin{cases}6\phi(x,t)+1, & \text{if }\phi(x,t) < 0 \\ 2\phi(x,t), ...
2
votes
1answer
67 views

Mathematical notation to define a function/relation

I have to define a function using mathematical notation. A function is translating one molecule ID to a vector(array,one dimensional matrix) of states that it can be in. I basically need to say the ...
0
votes
1answer
52 views

function symmetric around a point

I need some quick help solving this: What is y(ln(2))if the function y satisfies $$\frac{dy}{dx}=1-y^2$$ and is symmetric about the point (ln(4),0)? I know that a function is symmetric about the ...
2
votes
2answers
91 views

Evaluate the limit $\lim_{x\rightarrow 0} \frac{\sqrt{1-\sin(5x)}-\sqrt{1+\sin(5x)}}{x^2+x}$

Trying to find $$\lim_{x\rightarrow 0} \dfrac{\sqrt{1-\sin(5x)}-\sqrt{1+\sin(5x)}}{x^2+x}=\lim_{x\rightarrow 0} ...
1
vote
1answer
145 views

Inverse function, power set.

How to prove, that for every function $F: P(\mathbb N) \rightarrow P(\mathbb N)$, where: $F(\mathbb N)=\mathbb N$ $F(\emptyset)=\emptyset$ $F(\bigcup \Xi)=\bigcup\{F(X)|X\in\Xi\}$ for every $\ ...
0
votes
1answer
51 views

How to take derivative when the function is also parameterized

Why $$ \frac{\partial f_\tau(X_\tau)}{\partial \tau} = \left .\frac{\partial f_\tau}{\partial \tau}\right|_{X = X_\tau} + f'_\tau(X_\tau) \frac{\partial X_\tau}{\partial \tau} $$ where $f_\tau$ is a ...
1
vote
0answers
100 views

Is the Convolution of a Schwartz Function with an $ L^{p} $-Function a Smooth $ L^{p} $-Function?

Let $ n \in \mathbb{N} $ and $ p \in \mathbb{R}_{\geq 1} $. If $ f \in \mathscr{S}(\mathbb{R}^{n}) $ and $ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the ...
1
vote
1answer
92 views

How do I prove that $\cos(\frac{1}{2}x)$ is a periodic function?

Given: $f(x)=\cos(\frac{1}{2}x)$. Prove: f is a periodic function with period 4π My math teacher never went over this so I don't know where to start or what to do :/
3
votes
1answer
50 views

Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$

Let $f:[0,1]\to\mathbb{R}$ be continuous and $f(0)=f(1)$. Show that for any integer $n\geqslant 2$, there is $\xi\in(0,1)$ s.t.$f(\xi)=f\left(\xi+\frac{1}{n}\right).$ I think this requires the ...
3
votes
1answer
85 views

Why is the concept of a codomain useful?

I don't understand what the point is of specifying the codomain of a function. For example, if I ask, "Given the function f: $\Bbb R$ $\to$ $\Bbb R$, where f(x) = x^2, what is the image of f?", how is ...
3
votes
4answers
185 views

function with zero first to n'th derivative at end points

As an extension of my earlier question, It is required to find f(x) with following properties, $$ f(0) = 0 \hspace{1cm}f(1) = 1 \\ f'(0) = 0 \hspace{1cm}f'(1) = 0 \\ f''(0) = 0 \hspace{1cm}f''(1) = ...
1
vote
1answer
120 views

Graphic of the function: $\sin\left(\frac{5x^2+1}{x^4+1}\right)$

I'm trying to draw the graph of this function $\sin(\frac{5x^2+1}{x^4+1})$, but after the intersection with the axes and having made the derivative, I should find points of maximum and minimum, as ...
0
votes
0answers
56 views

Recursive function

Having difficulty with a question, was hoping someone could take a look and explain (if) where i'm going wrong. Consider the following recursive definition of a function $f:N\to N$ Base case: For ...
2
votes
1answer
122 views

Best way to explain how the Infimum and Supremum of this function are obtained…?

I have the function $\;f(x)=\dfrac{x^{(1/2)}}{2+x}\;$ and I know that $\inf(f)$ does not exist and $\sup(f)=2$ but I don't know how to formally show this rigorously? Anyone got a formal way of showing ...
1
vote
1answer
86 views

Why $xyz = e^x$ can be seen as the level surface $f(x,y,z) = xyz - e^x$?

That does not make sense to me. I recognize a level surface from the form $f(x,y,z) = k$. Where is the $k$ there? It looks just like a $3$ variables function to me.
2
votes
1answer
527 views

How to prove Cauchy Criterion for limits

Let $A$ be a nonempty subset of $\mathbb R$ and $f: A\rightarrow \mathbb R$. Suppose $c$ is a cluster point of $A$. Suppose the limit of $f(x)$ at $c$ does not exist. Show that there exists ...
1
vote
2answers
86 views

Determining function from simple xy graph

I'm not sure if this question is too vague for here. Will delete if necessary. I have a set of points, (1,782), (2,893), (3,992),... and I'm trying to determine the function that was used to generate ...
0
votes
1answer
153 views

Cantor-Lebesgue function and an increasing function are equal almost everywhere

Denote by $\varphi$ the cantor-lebesgue function and suppose $f$ is a certain increasing function defined on [0,1] and such that $f(x)=\varphi (x)$ for all $x\in[0,1]-C$ where $C$ is the cantor set. ...
3
votes
1answer
111 views

Functions that satisfy $f(x,z) = f(x,y) f(y,z)$

I am specifically looking for solutions that are NOT of the form: $f(x,y) = g(x)/g(y)$ since that is an obvious solution, as is $f(x,y) = 0$. I have a suspicion that there may be answers to this ...
2
votes
2answers
348 views

Prove that the greatest integer function: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$

Statement: the greatest integer function int: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$ Proof: let $x \in \mathbb{R}$, then $int(x) \leq x$ and is an element of $\mathbb{Z}$. Since ...
2
votes
0answers
97 views

How many tangents to the curve $y=x^3-px$ , can be drawn from different points in the plane?

$y=x^3-px$ so as ussual we find the slope of any tangent to the curve at a point $x_0$ $\dfrac{dy}{dx}=3x^2-p$ $3x_0^2-p$ so the equation of the tangent is ...
0
votes
1answer
85 views

Function from A to A

Let $S = \{1,2,3,4,5\}$ how many functions from $S \to S$ and how many of these are $bijective$ Say we also had $A = \{1,2,3,4,5\}$ then the number of functions would be $B^A$ and the number of ...
0
votes
4answers
356 views

Is it countable?

Show that the set of all bit strings (strings of 0’s and 1’s) is countable. Would you start by making a grid? I'm not exactly sure how to go about doing this?
2
votes
0answers
60 views

finding a function with given boundary conditions

$f(x)$ is given in the interval (a,b) with following properties, $$f(a) = 0 \hspace{1cm} f'(a) = 0 \hspace{1cm} f''(a) = 0 \\f(b) = 1 \hspace{1cm} f'(b) = 0 \hspace{1cm} f''(b) = 0 $$ I need to ...
1
vote
2answers
55 views

Countability (show set is countable)

Show that the set $\mathbb{Z}_+\times\mathbb{Z}_+$ is countable.? To solve this you have to show a one to one correspondence. $\mathbb{Z}_+\times\mathbb{Z}_+\to\mathbb{Z}_+$ Then my book recommends ...
2
votes
2answers
116 views

Prove cardinality

Let $V = \{x \in \mathbb{R} | 2 < x < 5\}$. Prove that $S$ and $V$ have the same cardinality, where $S$ denotes the set of real numbers between $0$ and $1$. The part I don't get is where my ...
2
votes
1answer
93 views

Difficult Discrete/Probability Problem

Here's the question: For a function $f:[n]\rightarrow[n]$, where $n$ is the set $\{1,2,3,\ldots,n\}$, define the inverse complexity, $ic(f)$ as the number of ordered pairs $\langle i,j \rangle$ ...