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

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2
votes
3answers
365 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 $f(a,...
0
votes
1answer
183 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
181 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
69 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 ...
3
votes
3answers
179 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 $10N$. I tried to find the minimum ...
1
vote
1answer
92 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 ...
1
vote
4answers
61 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 to ...
3
votes
1answer
79 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
62 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 ...
2
votes
1answer
37 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
84 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
52 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
59 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
78 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 \{1,...
0
votes
1answer
27 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}, y_{...
0
votes
1answer
29 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
143 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
66 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
77 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 $f(f(x_1)) = f(f(x_2)) = y$ ($f\...
0
votes
1answer
14 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
38 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, $f(5x)=2f(...
5
votes
1answer
64 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
83 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 c+b}...
1
vote
3answers
129 views

If a function has an inverse then it is bijective?

I have some trouble finding the answer to this, can someone help me out: If I have a general function $f$ with domain $X$ and codomain $Y$, I know nothing about the function (injective, surjective). ...
7
votes
1answer
153 views

Find all functions f with the following two properties

Let $f(x): [0, +\infty)\mapsto \mathbb{R}$ be a function such that for one $k\in [0, +\infty)$: $$f^2(x)=k^2+x\cdot f(x+k) \quad \forall x\in \{\;[0, +\infty) : x\geq k\;\}\qquad (1)$$ and $$\frac{...
0
votes
1answer
41 views

If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
0
votes
1answer
46 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
-1
votes
1answer
68 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [closed]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
2
votes
1answer
77 views

Solving functional equation gives incorrect function

Let $f:\mathbb{R} \to \mathbb{R}$ be a function which satisfies $e^xf(y)+e^yf(x)=2e^{x+y}-e^{x-y}$ for all real x and y. If I place $x=y$, I get $f(x)=e^x-\frac{1}{2}e^{-x}$ which does not satisfy the ...
1
vote
1answer
31 views

Partial vs. complete definition of a function

Suppose I define a function as $f(x,y)=2(x+y)$. Compare that definition to $f:\mathbb{R}^{2}\rightarrow\mathbb{R}, f(x,y)=2(x+y)$, which also gives the (co-)domain. Is there any standard way to refer ...
1
vote
2answers
64 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ \sum_{r=1}^{n}\frac{r^2(\...
3
votes
2answers
669 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
1
vote
3answers
184 views

Create a function where $f(1000)=1.99$ and where $f(400000)=0.49$

As per object I need to create a continuous and decreasing function with one variable where result is as above, but not only. Function should also be easy to modify to get $1<f(1000)<10$ and $...
2
votes
3answers
237 views

Consider the function f(x)=sin(x) in the interval x=[π/4,7π/4]. The number and location(s) of the local minima of this function are?

This is MCQ of a competitive exam(GATE), Answer is (d) given by GATE , and from other sources ,explanation is (b) somewhere and (d) somewhere , I am going with (b) as minimum at $270$, I have drawn ...
0
votes
0answers
38 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? $...
1
vote
0answers
66 views

Hieroglyphic from Herschel to Babbage?

John Herschel sent a letter to Charles Babbage in which he included this hieroglyphic with the message "Interpret it, it contains a great discovery". Personally I have no clue what it could mean. ...
0
votes
1answer
31 views

Finding the functions for circular reflection and their inverted forms

I'm trying to solve this exercise: Show that the transformation of inversion in the unit circle is given analytically by the equations $$x'=\frac{x}{x^2+y^2}, y'=\frac{y}{x^2+y^2}$$Find the ...
1
vote
1answer
71 views

Example of $f:\mathbb{R}\to\mathbb{R}$ injective and bounded, but with inverse not bounded or injective.

I am trying to come up with an example of a bounded and injective function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}$ is not injective or bounded. What are examples that could apply in this ...
7
votes
5answers
446 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
2
votes
0answers
70 views

Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. (...
0
votes
2answers
65 views

Prove that an equation has solution in R

Let $f:\mathbb R\to \mathbb R$, $x\in\mathbb R$ and $$f(x^2 + 3x + 1) = f^2(x) + 3f(x) + 1.$$ Prove that $f(x)=x$ has a solution $\in \mathbb R.$
1
vote
1answer
36 views

Non-linear system with functions

$f:\mathbb R\to \mathbb R$ monotonically increasing. Solve the system: $$\begin{cases} f(x) + x = f(y) + y\\ x^2 + xy + y^2 = 12\end{cases}$$ Since $f$ is monotonically increasing and $f:\...
1
vote
1answer
27 views

Can functions within a matrix adjust its size?

I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix: \begin{bmatrix}G\begin{...
0
votes
1answer
67 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
votes
1answer
54 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
0
votes
0answers
31 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
0
votes
2answers
62 views

function: bending the y=x line

My question has many relative questions but I didn't find anything exact to my needs. Let's take the function $f(x)=x$ with $x\in[0,100] $. I need to bend this and make it a curve. f will be a ...
1
vote
1answer
38 views

Show that the angle between $OP$ and the normal to the curve at $P$ satisfies the following

I'm struggling to answer the following question below I've already worked out the gradient to the curve at $P$, but I'm having difficulty answering the second part of the question. MY attempt is as ...
2
votes
0answers
58 views

How do you the roots of functions that are not quadratics?

I was asked to consider the equation $(x-3)(x+3)^2=c$ I have been asked to find the values of C in which the equation has: three distinct roots only one real root a double root and a single root ...
0
votes
1answer
74 views

algebra question.. [closed]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$