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

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2
votes
2answers
61 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
0
votes
1answer
16 views

Progressive Linear functions

I have a problem and I'm not sure how to calculate it or write it. It uses Linear functon as: y = am + b So the problem is that evrey time the unknown-m increases by 1, The unknown-a will add to ...
4
votes
4answers
111 views

Simple question about the pre-image of a set

Define the pre-image of a set $S \subseteq Y$ under $f$ where $f:X \to Y$ by $f^{-1}(S) = \{ x \in X : f(x) \in S \}$ Let $A = \{ 0 , 1\}, B = \{ 0,1,2,3 \}$. Define $f:A \to B$ by $f:x \mapsto x + ...
0
votes
0answers
8 views

Visible area of rotated plane?

Is there any general correlation between rotation and visible area of a plane? If the plane is perpendicular to my point of view I see exactly 100% of it. If it is rotated about the x-axis by 90° it ...
3
votes
2answers
85 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
0
votes
2answers
46 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
0
votes
1answer
46 views

Discrete Math: Functions and Set Questions

1) Consider the function: $f: \mathbb{R} \to \mathbb{R}$ (Real to Real Number), where $f(x)=2+x^2$, what would be all of the preimages of $3$? 1) $11$ 2) $11$, $-11$ 3) $1$, $-1$ 4) $1$ 2) Let $D ...
2
votes
1answer
38 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
1
vote
1answer
31 views

when is rational function regular?

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle $V(x^2+y^2-1) \subseteq A^2$is the rational function ...
1
vote
1answer
51 views

What's the name for a curve that takes the same amount of time to roll down no matter where you start on it?

In university I remember learning about a particular curve function with the unusual property that if you were to make a physical curved ramp out of it and roll a ball down it starting from rest, the ...
2
votes
1answer
46 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
-1
votes
1answer
53 views

is it possible to construct a function for Z x Z x Z -> N [closed]

is it possible to construct a function which satisfies Z x Z x Z -> N; where Z non-negative numbers, N is natural numbers.
1
vote
1answer
45 views

A one-to-one function from $\mathbb{Z}^+ \to (0,1)$?

I ran into a interesting homework question that asked me to find an example of a one-to-one function $f: \mathbb{Z}^+ \to (0,1)$. I'm thinking it should be some kind of linear function, or polynomial ...
1
vote
2answers
26 views

Function composition and differentiability

This problem asks for an example of functions $f$ and $g$ such that $g$ takes on all values, $f \circ g$ and $g$ are differentiable, but $f$ is not differentiable. I'm having trouble jumping straight ...
0
votes
2answers
30 views

define $f :R\to R$ by $f(x)=\frac{1}{(x-1)}$ when $x<1$ and $f(x)=\sqrt{(x-1)}$ when $x\geq 1$. Show that $f$ is a bijection and determine its inverse

A bonus Q on a discrete math/proofs test, I know I must prove injectivity and surjectivity, but am not exactly sure how to do so. Please help, this will be covered on the upcoming final exam in April. ...
-1
votes
1answer
49 views

Let $X \neq \emptyset$, define the relation$A\sim B$ if there exists a bijection $f : A \to B$, Show that $\sim$ is an equivalence relation on $X$.

A question on my last proofs midterm, I know I must prove injectivity and surjectivity, but there aren't really any obvious conditions or descriptions on S that helped me to manipulate it to try and ...
0
votes
1answer
36 views

Riemann-integrable functions and pointwise convergence

Hello, I was hoping for some advice on finding a function which will satisfy this. I think I am okay with the actual execution of the answer, but I don't know how I'm supposed to find a suitable ...
0
votes
1answer
15 views

Linear Growth Model

I have a problem where I have been given that $r(t)=at+b, 0 \leq t \leq \frac{100-b}{a}$. I have then been asked to find $t(r)$. Is this simply finding the inverse of $r(t)$?
0
votes
1answer
39 views

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ [closed]

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ Could anyone please help me? I dont have a clue how to start.
0
votes
1answer
25 views

Find pdf of $f(x)$ such that $g(x)/f(x)$ is approximately a constant

My friend asked me a question that asks to find a pdf function $f(x)$ such that $f(x)/g(x)$ is approximately a constant, where $g(x)=\sqrt{e^{x^2}+e^x}$, and $f(x) \neq g(x)$. And the range of x is ...
1
vote
1answer
14 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
0
votes
1answer
49 views

Spivak problem on property of continuous functions.

Ok so problem goes like this: If f is continuous on [0,1] and f(x) is in [0,1] for each x.Prove that f(x)=x for some x. My proof goes like this but I am not quite sure of my result. Let ...
1
vote
3answers
71 views

Showing a function $f$ has a zero

I have been working on this problem and was wondering if anyone could give me some hints as to how to answer this. So far as $p$ is a polynomial it is continuous and $g$ is given as being continuous, ...
1
vote
0answers
13 views

Moving function with vector (absolute values)

I have problem with functions and moving them with a vector. My first function is on image below Now I must draw graph of another function That's right. First, I must move function left 2 points ...
-1
votes
1answer
28 views

please prove Dom(h) = Dom(f)∪Dom(g)

Let f : A →B and g : C → D be two functions such that f(x) = g(x), ∀x∈A∩C. Then the union h of f and g defines the function h = f∪g : A∪C → B∪D where h(x) = f(x) if x∈A, h(x) = g(x) if x∈C . I ...
0
votes
0answers
19 views

Padé approximant of transfer function with gain and time delay.

$$ H(\omega) = A e^{-j \omega \tau} $$ I'm trying to use Padé approximation to generate a numerator and denominator polynomial for the above transfer function but genuinely struggling with how to ...
1
vote
2answers
27 views

How can I find the inverse of this function? [closed]

Can anyone help me find the inverse of this function? $$y=\frac{x}{2}-\frac{x^2}{16}$$
1
vote
1answer
28 views

A complex log question

I'm trying to find the solutions of $\log(z)=i\log(\bar{z})$ where $\bar{z}$ is the conjugate of $z$. I'm aware of the multivalued complex log, so $\log(z)=\log|z|+i\arg(z)$ but I don't see to be ...
0
votes
0answers
74 views

Solving sample size of hypergeometric distribution given a specific probability

I am trying to figure out how to calculate the sample size of a hypergeometric distribution, given a population, population successes, and probability. Here is the initial formula: ...
-1
votes
1answer
42 views

Python: matrix completion functions/libraries?

Are there functions in python that will fill out missing values in a matrix for you, by using collaborative filtering (ex. alternating minimization algorithm, etc). Or does one need to implement such ...
2
votes
2answers
49 views

interpreting limits

a short question: is this true that if: f(x) = $2^x$ g(x) = $100^\sqrt{x}$ $\lim\limits_{x \to \infty} \dfrac{f(x)}{g(x)}$ = $\infty$ then for x sufficiently large f(x) is always greater than ...
0
votes
1answer
9 views

Where a particular equation meets a particular axis

The question asks where the tangent plane to $z = e^{x - y}$ at $(1,1,1)$ meets the $z$-axis. Without performing any computations or even looking at the given function, based solely on the question ...
0
votes
0answers
14 views

Did I correctly apprehend this description of a binary function?

From Wikipedia, $f$ is a binary function if there exists $X,Y,Z$ such that $f: X × Y \mapsto Z$ .... ...one may represent a binary function as a subset of the Cartesian product $X × Y × Z$, ...
1
vote
4answers
42 views

How to show an injection?

I have to find an injection from $f:\mathbb{N}\times \mathbb{N} \to \mathbb{N}$ Basically, I cant imagine how this function might looks like. Do I have to find an injective function from ...
0
votes
0answers
10 views

How do Boolean-valued functions work?

Consider this function: P: X→ {true, false} There's nothing in that expression that says when X is true and when it is not true. How do these work?
3
votes
2answers
44 views

Domain and range of a function.

Find the domain and range of the function $$f(x)=\frac{1}{\sqrt{[\cos x]-[\sin x]}}$$ Where [] denotes the greatest integer function. I started as $[\cos x]-[\sin x]\gt0$ $\implies \cos ...
3
votes
1answer
48 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
-1
votes
3answers
36 views

Does the inverse of $g(x)$ exist? If so, what is the inverse, express in interval notation.

Let $g(x)=2+rx-3$ Does the inverse of $g(x)$ exist? Find the inverse of $g(x)$ and express in interval notation.
1
vote
1answer
60 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
4
votes
3answers
368 views

Growth rate of $n^{\sin n}$

Is there a way of comparing the growth of functions $ f(n) = n ^ {\sin(n)} $ and $ g(n) = n ^ {1/2} $ in terms of $ O, o, \Omega, \omega, \Theta $ ? Periodically, $ f(n) $ keeps going above and ...
1
vote
1answer
54 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
0
votes
0answers
19 views

Superadditivity ; How to prove Superadditive function?

How can I prove this superadditive function? It's quite confused
0
votes
1answer
22 views

Trigonometric functions over arbitrary angles

Trigonometric functions over obtuse or arbitrary angles doesn't make sense. We can only imagine for eg. sin(x) for angles < 90 degrees because it represents the ratio of the opposite and ...
2
votes
2answers
48 views

Composition with function whose graph is everywhere dense in $\mathbb R ^2$

Let $f\colon \mathbb R \to \mathbb R$ be a function whose graph is everywhere dense in $\mathbb R ^2$. Let $I\subseteq\mathbb R$ and $g\colon I\to \mathbb R$ be a surjective function. I'm concluding ...
0
votes
1answer
68 views

Let $S =\{1,2,3,4,5 \} $ For each give a brief explanation and simply answer to a number. (a) How many functions $f: S \longrightarrow S$ are there?

(b) How many one-to-one functions $f : S\longrightarrow S$ are there? (c) How many functions $f : S \longrightarrow S$ are there so that $f o f(1) = 2$ ? (d) How many onto functions $f: ...
2
votes
1answer
56 views

A question on Fourier Transform

Is there a function which is not absolutely integrable but which has a continuous fourier transform? I know that if a function is absolutely integrable then the fourier transform is continuous but I ...
0
votes
1answer
58 views

Prove that this function defined by f(x) is bijective.

Prove that the function $ f: \mathbb{R} - \{1\} \to \mathbb{R} - \{1\}$ defined by $ f(x) $ is bijective. $$ f(x)=\left({x+1\over x-1}\right)^3 $$ I am taking my first Computer Science ...
0
votes
3answers
63 views

Proving a function is a one to one correspondence

I understand that to show a function is a one to one correspondence, you have to show that the function is both one to one and onto. Proving a function is one to one seems simple enough. However for ...
0
votes
1answer
39 views

Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
3
votes
1answer
38 views

Discrete math functions help?

I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test? ...