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

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0
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1answer
18 views

Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where ...
0
votes
1answer
17 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
2
votes
1answer
33 views

Distribution for function

I would like a good book to study distribution or generalized functions like the "Basic idea" of that Wiki page. Is there anyone could give me some good book references in this domain? Thanks!
0
votes
0answers
121 views

What is the range of this function?

What is the range of $h$? $f(x)=4x+1$ $g(x)=(x-1)/3$ Let $h=\{f^n(g^m(1)):n,m\in\mathbb{N}\geq0\}$ What is the range of $h$? Show that $(2\mathbb{N}-1)\subset H$. ... okay I've done a bit more: ...
-1
votes
1answer
15 views

Limit through a figure

If a circular arc of radius 1 subtends an angle of x radians . The centre of the circle is o and the point c is the intersection of two tangents lines at a and b . Now let T(x) be the area of the ...
0
votes
3answers
27 views

Continous function

There are two functions g(x)= ($(2x+1)^{1/2}$-$1$)/x , where x is not equal to zero = 1 , x=0 h (x) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
2
votes
1answer
20 views

Determine the composition of the functions $f(x)=4x+3$ and $g(x)=-5x^2+1$

Answer: \begin{align*} (f \circ g)(x) & = f(g(x))\\ & = 4(-5x^2+1)+3\\ & = -20x^2+8+3\\ & = -20x^2+11 \end{align*} \begin{align*} ...
0
votes
1answer
13 views

Demonstrate that a function is periodic knowing symmetry axis.

Let $$f:\mathbb{R}\rightarrow \mathbb{R}$$ with symmetry axis at $x = {1,2}$. Demonstrate that the function is periodic.
-2
votes
1answer
68 views

The range of a twisted composition of sines [on hold]

Let $f(x) = \sin \left( \frac \pi 6 \sin \left( \frac \pi 2\sin x \right) \right)$ and $g(x) = \frac \pi 2 \sin x$ for all $x \in \Bbb R$. Which of the following are true? The range of $f$ is ...
12
votes
3answers
313 views

$f(nx)\to 0$ as $n\to+\infty$

Let $f:\mathbb R^+\to\mathbb R^+$ be a continuous function, and let $I$ be a subset of $\mathbb R^+$ such that the following property holds: For any $x\in I$, $f(nx)\to 0$ as $n\to+\infty$. ...
1
vote
1answer
22 views

What is the slowest growing function that is total but not primitive recursive?

For what I have in mind is the Ackermann-Buck function. If there isn't a slowest growing function do you have examples of other function slower growing than Ackermann-Buck's function?
-1
votes
1answer
31 views

If the function $f : \mathbb{R} \to A$ given by $f(x) = \frac{e^x - e^{-|x|}}{e^x+e^{|x|}}$ is a surjection, find $A$

If the function $f : \mathbb{R} \to A$ given by $f(x) = \dfrac{e^x - e^{-|x|}}{e^x+e^{|x|}}$ is a surjection, find $A$. I know the fact that "range = co-domain" but was not able to proceed.
-1
votes
0answers
18 views

use the definition of limits to find f(x), f(y) in a function with multiple variables [on hold]

I have the next function $f(x,y)$ = $(x^2+6xy+7y^2)/(6x^2+7y^2)^{(1/2)}$ and I need find the derivative of this by using the definition of limits and I have no Idea of how to start or develop this ...
2
votes
3answers
29 views

Determine the function to be injective when $f: A\to N$, where $A=\{1,4,3\}$ and $f(x)=x^{2}$.

I claimed the function is injective since the elements in domain maps uniquely to elements in the codomain when plug into $f(x)=x^{2}$ . which gives $\{1,16,9\}$ where $\{1,16,9\}$ is found in the ...
9
votes
2answers
1k views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
0
votes
0answers
21 views

How do I express this function image?

Problem says: if $f(x,y)= (x-y,x,y)$, find $f(\mathbb{R}^2)$. I can see the graphic (thanks to wolfram alpha), but I'm not sure what they are asking me to write with "$f(\mathbb{R}^2)=?$"
0
votes
1answer
18 views

ordered set notation in functions

Do please forgive me, if this question is a duplicate. How does one correctly notate a function $f$, which takes a ordered subset $S$ from the field $\mathbb{K}$ and returns an other (ordered) subset ...
0
votes
0answers
22 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
0
votes
1answer
35 views

Is this the correct way to translate this phrase into symbols?

The domain of g is the set of all real numbers $x$ such that $x$ is not equal to $-3$. $$g(x)=\{x:\mathbb R|x\ne -3\}$$
7
votes
3answers
99 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle ...
0
votes
2answers
22 views

Equal functions with non-equal definitions

Suppose we have two functions $f, g: \mathbb{D} \rightarrow \mathbb{R}$ where $\mathbb{D} \subseteq \mathbb{R}$. Also, $\forall x \in \mathbb{D}: f(x) = g(x)$. My question is: Is it possible for such ...
-1
votes
0answers
22 views

Periodicity of functions

What would be the period of $f(x)=n(n+1)x$. As the period of a constant function is not defined hence what would be the answer to this.
0
votes
1answer
39 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
0
votes
2answers
23 views

Proof about composed functions

Let $f\colon X \to Y$ and $g\colon Y \to X$ be functions. Assume $g \circ f$ is bijective. Prove $f$ is injective and $g$ is surjective. Approach: if $g \circ f$ is bijective then $g \circ ...
-4
votes
1answer
33 views

Introduction to set theory, Natural Numbers [on hold]

Proof that for any $n\in\mathbb{N}$, if $f:n\to n$ is a one-to-one function, then $f$ is surjective. (Where n={0,1,2,...,n-1} is a natural number) This is what I've done, but not if this good: We ...
0
votes
1answer
23 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
4
votes
2answers
176 views

Find the domain

I have been a bit confused about finding the domain of these functions. 1) $\dfrac{12}{2x+3}$ 2) $\dfrac{4x-3}{x^2-81}$ 3) $\dfrac{x^2 -3x -18}{x-6}$ So I solved for $x$ and then those were the ...
-1
votes
1answer
31 views

Prove that g:R$\to$ (-1,1] defined by g(x)= cos x is surjective or not?

Surjective: a function $f$ from $A$ to $B$ is called onto or surjective, if and only if every element $b \in B$ there is an element $a \in A$ with $f(a)=b$. Really appreciate if someone explain ...
-3
votes
2answers
50 views

How do I develop a formula to find the height of a male using his femur? [closed]

I'm a grade eleven math student and I need a formula to find the height of a 15/16 year old male subject using only the length of his femur. I'm very confused on how to do this, please help, it's for ...
1
vote
0answers
13 views

Show that $C/B \approx T/A$

Let $T=S^1 \times S^1$ be the torus with meridian $A=S^1 \times \{1\} \subset T$ Let $C=S^1 \times [-1, 1]$ be the cylinder with base circles $B=S^1 \times \{-1, 1\}$. Show that $C/B \approx ...
0
votes
0answers
18 views

If f:[0,∞)→R is a **continuous and bounded** function -> so f has a maximum, minimum or both

I need to find an example that contradicts this next sentence: If f:[0,∞)→R is a continuous and bounded function -> so f has a maximum, minimum or both Any tips about how to solve this question? ...
2
votes
2answers
33 views

Is a function of a function the same as a two-argument function?

Say I have a function $g$ described by $$(x\stackrel f\mapsto y)\stackrel g\mapsto z$$ That is, $g$ takes a function, $f$, and maps it to another object, maybe just a real number or vector or ...
0
votes
2answers
36 views

Relation between $\int_{a}^{b} f(x) dx$ and $\int_{a}^{b} (1-f(x)) dx$

Say you're expected to work out $\int_{0}^{\pi/3} \sin^2(x) dx$ solely from the result $\int_{0}^{\pi/3} \cos^2(x) dx$. It can be transformed into $\int_{0}^{\pi/3} (1-\cos^2(x)) dx$, but then what?
2
votes
1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
2
votes
2answers
36 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
0
votes
0answers
38 views

Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

How would one approach the following problem? Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole So I need a function $f(x,y) : ...
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votes
0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
4
votes
3answers
163 views

Why is $\log(1+e^x) - \frac{x}{2}$ even?

I'm dealing with Fourier series and I'm trying to figure out $\log(1+e^x) - \frac{x}{2}$ is even??? I've tried the $f(-x) = f(x)$ method but it doesn't give me the equality. But I've plotted it, and ...
1
vote
0answers
52 views

About a geometric algorithm to compute $\sin$ based on the unit circle

In an old post I have found a user which claims to have a geometric algorithm to compute trigonometric  functions for an angle between $0^\circ$ and $90^\circ$ based on the unit circle. Here's the ...
2
votes
6answers
530 views

Separation of variables

If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$? Thanks.
4
votes
4answers
550 views

How are domain and co-domain of a function useful?

I'm at university and I learned linear algebra, set theory, logic, and other kind of mathematics that use functions a lot. Now, I know that function is very important and useful in mathematics but I ...
2
votes
2answers
59 views

How to “rotate” a function? Or, how to write a function which has a known, rotational symmetry with respect to another function?

EDIT 2: I've posted my "real" question here: http://mathematica.stackexchange.com/questions/115766/finding-closed-form-eigenvalues-of-a-particular-matrix I have posed my question formally in LaTeX ...
1
vote
1answer
22 views

$L^2$ - base functions

say we have a function, $f$, in $L^2$ and have different base functions for $L^2$, then is there a reason to believe that one base is "better" compared to the other when writing expansions for $f$? ...
0
votes
1answer
135 views

Prove an additive function has property f(x)=x

So I am given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$, is continuous at $x=0$, and $f(1)=1$. I need to show that $f(x)=x$ for all real ...
0
votes
1answer
67 views

Create an equation for a description of a rational function

A graph has a $y$-intercept at $-5$, no $x$-intercepts, and discontinuous points at $(-1,-5)$ and $(3, -5)$. I want to form an equation for this graph, but I don't know how the $y$-intercept relates ...
3
votes
1answer
51 views

What might this function be?

The problem: I'm looking for a particular function $f(x, y)$—this isn't "homework" in the sense that I have no idea if such a function exists. It has a continuous domain $-1 \lt x \lt 1$ and $-1 \lt ...
1
vote
0answers
45 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
1
vote
1answer
46 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
0
votes
1answer
40 views

Continuous function, finding its value [duplicate]

If a function $f: \mathbb{R}\to\mathbb{R}$ is continuous and $f(x+y) = f(x) + f(y)$ for all $x,y\in\mathbb{R}$, then what is this function $f(x)$?
0
votes
1answer
59 views

Finding value of functions $f(x) g(x)$ [closed]

If $f(x)=2x^3+4x^2+3x+2$ and $g(x)=2x^3+x^2+4$, where $f(x), g(x) \in \mathbb{Z}_5[x]$ then $f(x) g(x)$ is equal to ?