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

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2
votes
4answers
83 views

If $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function?

Like the title reads, if $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function? Assume $x,y,\alpha\in\mathbb{R}$.
0
votes
0answers
14 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
1
vote
1answer
59 views

Function for this series of numbers

I have $f(n)$: $f\left(1\right)=1$ $f\left(2\right)=2$ $f\left(3\right)=3$ $f\left(4\right)=3$ How can I solve the next $f(n)$? I came to this problem while been working on my algorithm. Here is ...
0
votes
3answers
53 views

The range of the function $f(x)=\frac{\sin x}{\sqrt{1+\tan^2x}}-\frac{\cos x}{\sqrt{1+\cot^2x}}$

Find the range of the function $f(x)=\frac{\sin x}{\sqrt{1+\tan^2x}}-\frac{\cos x}{\sqrt{1+\cot^2x}}$. By simply looking at the problem and simplifying trigonometrically,it looks as if range is zero ...
8
votes
1answer
85 views

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$

Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$ $(A)\frac{1}{10}\hspace{1 cm}(B)10\hspace{1 cm}(C)1\hspace{1 cm}(D)$cant be obtained I could not solve this question.I ...
0
votes
1answer
35 views

If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$…

Question: If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$, then $f(f(x))+f(f(\frac{1}{x})) = ?$ (a)p (b)2p (c)-p (d)-2p Attempt: ...
1
vote
3answers
25 views

Basic question about showing bijection when a function is defined piece wise

Hello there everyone, I was told that the standard proof for showing that $$| \mathbb{N} | = | \mathbb{Z} |$$ is to define $$f: \mathbb{Z} \to \mathbb{N}$$ as $$f(x)= \begin{cases} 2x &\text{ ...
-1
votes
1answer
59 views

How to sketch a function which is 2 to the power of x [closed]

Hi I'm looking for help to sketch a function. $$ (e)\quad Sketch\quad the\quad graphs\quad of\quad each\quad of\quad the\quad following\quad functions:\\ (i)\quad f:\mathbb{R}\rightarrow ...
0
votes
1answer
29 views

Which functions can be represented as matrices?

I was reading the intuition for the associativity of matrix multiplication and it was given to be analogical to composition of functions. So which functions can be represented as matrices and how?
0
votes
1answer
53 views

counter examples for Riemann and Lebesgue integrabilities

I'm seeking interesting examples that are not mentioned in usual real analysis texts. It seems that in general there is no relationship between Riemann integrability and Lebesgue integrability when ...
1
vote
0answers
25 views

About an asymptote of Golomb's sequence and an asymptote of a sequence in general

In this question, I asked for the solutions to the differential equation $f(f(x))=\frac{1}{f'(x)}$ because I think that a solution to this equation is related to Golomb's sequence. Let me explain, ...
1
vote
1answer
64 views

How to solve this inequality question?

$abc=1$ where $a$, $b$, $c$ are positive reals. Prove that $$\sqrt{\frac{a}{a+8}} + \sqrt{\frac{b}{b+8}} + \sqrt{\frac{c}{c+8}} \ge 1$$
0
votes
2answers
38 views

Existence of (nearly) number of digits preserving pairing function?

A pairing function $\pi$ is a bijective mapping $ℕ×ℕ → ℕ$. That means it can reversibly encode two integers $a$ and $b$ into one integer $p$. The most well known pairing function is the Cantor ...
-4
votes
2answers
56 views

Odd and even functions [closed]

For the statement below, either prove it is true, or find a counter example to demonstrate that it is false. If $f$ and $g$ are both odd functions defined for all real numbers, then $f(g(x))$ is ...
2
votes
1answer
47 views

Find $a$, given $y(n)=x(n)+ax(n-d)$, interesting question

Me and two friends of mine are working on a project (scholarly purposes only). The goal of this project is to clean an audio signal (speech, a song, anything audio) of echo. Generally speaking, if ...
-2
votes
1answer
24 views

Composite functions of f and g [closed]

Prove if it is true, or provide a counter example if it is false. For every functions $f$ and $g$ we have $f\big(g(x)\big) = g\big(f(x)\big)$.
0
votes
2answers
93 views

Proving there exist an interval and a number $p$ where $f(x) \leq x^p$ holds

We have a function $f:\mathbb{R}\to [0,1]$, where $f(x)=0$, for $x\leq 0$, and $f$ is a right-continuous function. How can we prove that there exist a number $0 \lt p \leq 1$ and a number $0\lt a$, ...
0
votes
0answers
34 views

Norm inequality that i cannot show

I have recently stumbled upon an inequality that i can't show \begin{align} \int_a^b\|f(t)-f(b)\|^2dt\leq(b-a)^2\int_a^b\|f'(t)\|^2dt \end{align} None of the tools I know handle the derivative on the ...
1
vote
0answers
23 views

Notation of functional relationships in Fuzzy Logic

I would like to know how to put the following ideas in to a more formal mathematical notation: Traditional methods implement a direct functional relationship between input and output signals: ...
0
votes
1answer
27 views

Sketching a complicated function

How does one sketch $t^2\sin(\frac{\pi}{t})$ without using graphing devices? I know that we could use the method of sketching separate points and then connecting them, but I don't think that this ...
14
votes
6answers
433 views

Is there a continuous function $f(x)$ such that the inverse function is $1/f(x)$?

A student came to me with this question and I cracked my head for one hour but I couldn't unambiguously prove that it exists or it doesn't exist. Continuous and invertible function such that ...
2
votes
1answer
33 views

If $\ne: X \times X \to S$ is continuous, is X hausdorff?

The Sierpiński space is defined like so: $$S = (\{\top, \bot\}, \{\emptyset, \{\top\}, \{\top, \bot\}\})$$ (A nice way to visualize is to take [0, 1], and glue 0 on $\bot$ and (0,1] onto $\top$.) ...
2
votes
0answers
65 views

For any function $f(xy) = f(x) f(y)$

I'm stuck on a homework question and would really appreciate any help. I have to prove or find a counterexample on the following statement: For any function $f(xy) = f(x)*f(y)$. I'd say this isn't ...
0
votes
1answer
28 views

Is $f: (0,\pi/2)\to [0,1]^2/ f(x)=(\cos x, \sin x) $ injective?

Is $f: (0,\pi/2)\to [0,1]^2/ f(x)=(\cos x, \sin x) $ injective? I set $f(x)=f(y)\implies $ $\cos x=\cos y$ and $\sin x=\sin y$. So applying the inverse function to both should result in $x=y$, but ...
2
votes
2answers
120 views

Prove that $\varphi^{-1}$ is the inverse of $\varphi$

This is exercise 2.27 of Lee's introduction to topological manifolds. I proved (geometrically) that $$\varphi(x,y,z)=\frac{(x,y,z)}{\sqrt{x^2+y^2+z^2}}$$ and that ...
2
votes
1answer
23 views

$A:X \to Y$- linear continuous function, $A\neq 0$ $H=\ker A\implies \exists z \in H^{\bot}: \|z\|=1$

$A:X \to Y$- linear continuous function, $A\neq 0$ $H=\ker A, X=H\oplus H^{\bot}\implies \exists z \neq 0 \in H^{\bot}: \|z\|=1$ This implication is confusing to me, here in the book it says in ...
2
votes
3answers
134 views

Formula for alternating sequences

I am looking for a general formula for alternating sequences. I know that the formula $f(x)=(-1)^x$ gives the sequence $1,-1,1,-1,...$ but I want more a general formula; for example the function ...
0
votes
1answer
26 views

How convert a scalar to a vector?

I know that if we have the function $$ y(x)=\frac{k}{x}, $$ then the vector form of this is $$ \vec{y}(\vec{x})=\frac{k}{x^{2}}\vec{x}, $$ where $\vec{x}$ is a vector Does anyone know how to ...
1
vote
5answers
127 views

Why is $f(x)=\sqrt x $ not a function?

Why is $f(x)=\sqrt x$ not a function? I understand that the definition of a function states that every "input" must be related to exactly one "output", but I am curious as to the WHY.
1
vote
3answers
52 views

Continuity of the inverse map

If we have a function $F(x): \mathbb{R^4} \rightarrow \mathbb{R^3}$. Defined as \begin{align} x_1\, x_4&=y_1 \\ x_2\, x_4&=y_2 \\ x_1^2+x_2^2-x_3^2&=y_3 \end{align} Can a continuous ...
1
vote
2answers
49 views

The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$

The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$ contains $N$ integers. Find the value of $10N$. We have to find number of ...
-1
votes
0answers
30 views

How to analytically describe the number of peaks on a roughness surface?

I have a question about how to model the number of peaks over a length $l$ on a surface, which is defined as: Let $f$ be the profile of the roughness of a surface. Let $c_1$ and $c_2$ be constant ...
1
vote
1answer
31 views

The derivative of piecewise-defined function gives undefined in knot point

I have defined a (rather simple) piecewise-defined function in Maple. But when I am finding the derivative to the piecewise-defined function, it suddenly sets the 'knot point' as undefined even ...
3
votes
4answers
53 views

$f(x,y)=(\sin x e^y, \cos x e^y)$- this is continuous. How would I prove that it's inverse is continuous as well?

$f(x,y)=(\sin (x) e^y, \cos (x) e^y)$- this is continuous. How would I prove that it's inverse is continuous as well? I need this for inverse function differentiating theorem that says that f has ...
3
votes
1answer
42 views

Integration: The Periodic Function and its Properties

I am aware that if $f(x)$ is a periodic function with period $T$ then: 1.) $\int^{nT}_{0}f(x)\,dx = n\int^{T}_{0}f(x)\,dx$, for $n$ an integer. 2.) $\int^{T+a}_{a}f(x)\,dx = \int^{T}_{0}f(x)\,dx$, ...
0
votes
2answers
52 views

Show that $f(p) + f(-p) = 2f(p^2)$?

So I have a functions question.. again. $f(x) \rightarrow \frac{2}{x-1}$ and $x\neq{}-\frac{b}{a}$ Show that $f(p) + f(-p) = 2f(p^2)$. My Work: $\frac{2}{x-1}+\frac{2}{-x-1}$ which is ...
3
votes
1answer
32 views

$ \text{If } f,g \in D(U) \implies \alpha f + \beta g \in D(U)(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$

Prove: $f,g$ are differentiable functions on open set $U \implies \alpha f + \beta g$ is differentiable on $U$ as well. Furthermore, $(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$. Proof: We ...
0
votes
1answer
61 views

Find twice continuously differentiable bounded function vanishing at infinity satisfying f(0)=0 and f'(0)=0?

I am looking for a twice continuously differentiable and bounded function (i.e $f$, $f'$, and $f''$ bounded) vanishing at infinity $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$ and $f'(0)=0$ . Without the ...
0
votes
2answers
42 views

Prove that function is monotonic

I have two functions for which I have to prove that they are monotonic for $x\in (-\infty,0]$. The first function is: $f(x)=\frac{1}{2}\left( 2+x^2-\sqrt{4x^2+x^4} \right)$, the second function is ...
2
votes
3answers
54 views

Range of function $g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}}$

If range of $ f(x)$ is$[-1,1]$,then what is the range of function $g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}}$? My attempt:As $-1\leq f(x)\leq 1\Rightarrow 0\leq |f(x)|\leq 1 $ Therefore ...
4
votes
2answers
62 views

Function Can Only be Solved by Simultaneous Equations, returns different/wrong answer each time it is solved?

I'm having a lot of trouble with a specific question regarding functions, but I'm not sure where to post it.. the question is: Let $$ y = f(x) = a x^2 + bx + c $$ and have the values ($i \in ...
0
votes
1answer
24 views

Generating function from a set of arbitrary integer values

Is it possible to define a function defined in domain $\mathbb{R}$ which curves through a set of points $(n, a_n)$ where the function's local minimum and maximum between $(x_i, y_i)$ and $(x_{i+1}, ...
0
votes
1answer
14 views

Which function is suitable to approximate a convex piecewise linear function

Im trying to fit a convex piecewise linear function into a smooth function. However I have no idea which kind of function is suitable? Can anyone give me some examples of function that is suitable ...
3
votes
4answers
119 views

Is there a purpose behind a function?

As I understand it, a function is a relation between two sets of numbers where as for every input value there is only assigned one output. Or for every $x$ there is only one $y$. What I don't ...
1
vote
1answer
43 views

What can be said about the continuous function $f:\mathbb R^{2} \rightarrow \mathbb R$ that has only finitely many $0$'s $?$

$f\colon \mathbb R^{2}\rightarrow \mathbb R$ is a continuous map that assumes $0$ for only finitely many points. Then which one is true A. either $f(x)\le 0$ for all $x$ or $f(x)\ge ...
1
vote
2answers
53 views

Prove that f is injective. (Is my solution correct?)

Let $f: R\to R$, and ($f\circ f\circ f)(x) = (f\circ f)(x) + x$, $x$ $\in \mathbb R.$ Prove that $f$ is injective. My Solution: Let $x_1, x_2\in \mathbb R.$ and $f(f(x_1)) = f(f(x_2)) = y$ ...
0
votes
1answer
13 views

On determining a function given a certain parametrization of a point

Imagine we have a parametrization of a particle in 2D space like this http://i.minus.com/iXL64EfdJe6w5.gif How do we go about finding an explicit way to express these functions ($f(x)$ and $g(x)$) ...
0
votes
1answer
36 views

$f(x)=2-|x-3|, 1\le x\le 5$ and for other values, $f(x)$ is obtained using the relation $f(5x)=kf(x)$ for $x\in R$. then…

Question: The maximum value of f(x) in $[5^4,5^5]$ for $k=2$ is? Also, if $$\lim_{x\to \infty}\int_1^xf(x)dx$$ is a finite number, find the exhaustive set of $k$. Attempt : For first part, ...
6
votes
1answer
54 views

Are custom named functions acceptable notation?

A custom name being, for example, my function name (MFN): $MFN(x) := ax + b$ As contrasted with: $\delta(x) := ax + b$ Questions: Is it permissible to name the function $MFN$ above? Or is this ...
0
votes
1answer
38 views

Tight bounds for Bowers array notation

This link http://googology.wikia.com/wiki/Array_notation shows the definition of bowers linear array notation and the approximation $$\{n,a+1,b+1,c+1,d+1,...\}\ \approx f^a_{...+\omega^2d+\omega ...