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

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-5
votes
0answers
27 views

Problem about the domain of a function [closed]

The problem $f(x)= \begin{cases} \dfrac{ x + 8 }{ ( x + 8 )( x - 9 ) } & \text{if } x \neq 9 \\ 1 & \text{if } x = 9 \end{cases}$ $f(x)= ​⎩ ​⎨ ​⎧ ​​ ​ ​(x+8)(x−9) ​ ​x+8 ​​ ​1 ​​ ​if ...
0
votes
0answers
45 views

How to find $k$ such that the equation possesses only one real root?

How to find $k$ such that the equation possesses only one real root ? $$f (x)=e^x-1-k\tan^{-1} (x)$$ I've plotted $e^x-1$ and $\arctan(x)$ Now it can be visualized that for $k<0$ and $k=1$ ...
1
vote
1answer
81 views

Find real $x$ satisfying $2^x + 2^{|x|} ≥ 2\sqrt{2}$

Solve for $ x\in \mathbb{R}$ if $$2^x + 2^{|x|} \ge 2\sqrt{2}$$ The answer is given as $\big(-\infty,\log_{2}(\sqrt{2}-1)\big] \cup \left[\frac 12, \infty\right)$.
0
votes
2answers
32 views

Square Inequalities

Are the following two statements both true? (maybe there is some small exception to one of them?) A) Let f, g be real-valued functions. Then, the following is true for all such functions: ...
-4
votes
1answer
36 views

Find the value of $f(1,100)$ [closed]

Let $f$ be defined on $$\{(i,j):i\; and\; j\; are\; positive \;integers\}$$ satisfying $(i) f(i,i+1)=\frac 13, \forall i$ $(ii) f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j), \forall k$ such that $i\lt k\lt j$ ...
0
votes
2answers
41 views

reflecting a function about a line

Say you have a function $f(x)$ and a line $g(x)=ax+b$. How do you reflect $f$ about $g$? I am apparently supposed to write more text, but the line above is all I am after, hence I wrote this sentence ...
2
votes
1answer
68 views

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$?

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$? It's obviously false. But the point is that "can it be proved without using the fact that the ...
2
votes
2answers
37 views

Prove a finite limit based on a given infinite limit

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that $$\lim_\limits{x\to 3}{\frac{f(x)+1}{f(x)-1}}=+\infty$$ with $f(x)\neq 1$ close to $x_0=3$. Prove that: a) $\lim_\limits{x\to 3}{f(x)}=1$ ...
-2
votes
2answers
30 views

specifying an asymptotic function

Please help me out; I need to specify a function satisfying these conditions: $$ f(0)=1 \;\;;\;\lim_{x \to \infty}f(x)=0$$ Hopefully does there exist a simple answer? Thanks a lot!
0
votes
1answer
27 views

Function equation bound between rectangle

I need to specify the equation form. I have 4 known points then the equation is: $$f(x)=ax^3+bx^2+cx+d.$$ I can solve and find $a, b, c, d$. But also I want this equation to be bound between two ...
1
vote
2answers
38 views

An injection between finite sets of equal size must be a bijection

To me, it seems logical that if I have two finite sets of equal size, and there is an injection between them, then that injection must be a bijection. However, of course, we cannot just claim these ...
-1
votes
2answers
27 views

Vector parametrization of a surface intersection

How does one parametrize the following curve in 3-space to $\vec{g}(t): [a, b] \to \mathbb{R}^3$: the intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$ ? What I could come up with is as follows: ...
0
votes
1answer
39 views

How do I show this function is monotonically decreasing?

Let $p = $Probability of head on a coin toss ; $p < 0.5$ (biased coin). $f(k) =$ Probability that heads is the majority from $k$ tosses, where $k$ takes odd values. I want to show that $f(k)$ ...
0
votes
2answers
105 views

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$ (I know the inverse of a closed set of a continuous function is closed, but is this a must?) And does the following apply to all ...
0
votes
3answers
32 views

How to set a function value or expression over a domain in Maple

I have a function $f(x,y,z)$ and would like to impose the condition $$f(x,y,z)|_{x^2+y^2<1}=x-y$$ That is, $f$ is set to be equal to $x-y$ within the unit circle in the xy-plane. How would I ...
-2
votes
1answer
29 views

Determine values of $a$ given $c$ in quadratic equation and a point $(1,2)$ [closed]

The point $(1,2)$ is on the graph of the quadratic function $f(x) = ax^2 + bx + 1$. Determine the values of $a$, such that the graph of $f(x)$ intersects the $x$-axis at two distinct points. This ...
10
votes
4answers
103 views

How can you factor $x^4-6x^3+8x^2+2x-1?$

The original question is: Solve this equation for x: $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ I expanded and simplified it to get $$x^4-6x^3+8x^2+2x-1=0$$ Since neither -1 nor 1 are factors, it appears ...
0
votes
3answers
45 views

Function that returns $0$ for all negative values

I need a function that returns $0$ if the given number is negative, and otherwise doesn't change the number. Example: $$y(-5)=0,\ y(-2)=0,\ y(0)=0,\ y(3)=3,\ y(2566)=2566.$$ Does such a function ...
-2
votes
0answers
30 views

Creating a utlity function from fuzzy UTASTAR method [closed]

I have difficulty in understanding equations in research paper. The research paper is about creating utility function from fuzzy data. The method presented in paper is called fuzzy UTASTAR. The ...
0
votes
1answer
29 views

Varying definitions for concavity of a function

I've been working on concavity of functions and have noticed that different texts define this notion in different ways. Specifically, some texts include the endpoints of an interval when describing ...
0
votes
1answer
43 views

Find Domain and Range of implicit function $ $x$ $y$^2 -$x$^2$y$ + $x$+$y$ = 2 $

I have this function. I noticed that it can be written as: $ $x$ $y$^2 +$y$(1-$x$^2) + ($x$ - 2) = 0 $, so this is a quadratic in y. Thus \begin{equation} y=\frac{(x^2-1)\pm \sqrt[2]{(x^2-1)^2 ...
3
votes
1answer
56 views

Is $f(x,y)=ax^2+by^2, \ a,b \in \mathbb R $ a bijection between $\mathbb R^2 \to \mathbb R$? Bijections of topologies

Is $f(x,y)=ax^2+by^2$ a bijection between $\mathbb R^2 \to \mathbb R$ ? How about $f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2}+ \frac{z^2}{c^2}? ( \mathbb R^3 \to \mathbb R )$ What confuses me now ...
0
votes
3answers
64 views

If a mapping $g: \mathbb{R} \to \mathbb{R}$ is strictly increasing, is it an open map?

If a mapping $g: \mathbb{R} \to \mathbb{R}$ is strictly increasing, is it necessarily an open map? i.e. for $a,b \in \mathbb{R}$ and $a<b$ can we conclude that ...
0
votes
1answer
51 views

Bowers array notation : $f_{\omega^\omega}(n)\approx [n,…,n]$ ($n$ times)

I learnt at this site that $$\large f_{\omega^\omega}(n)\approx \underbrace{[n,...,n]}_{n\ n's}$$ For a simular approximation $$\large f_{\omega^2}(n)\approx \underbrace{n\rightarrow ...
11
votes
4answers
730 views

When do two functions become equal?

When do two functions become equal? I have stumbled over this definition of equality of functions in elementary real analysis. Let $X$ and $Y$ be two sets. Let $f:X\rightarrow Y$ and ...
6
votes
6answers
410 views

Prove that the coefficients of a quadratic function with real roots cannot be in geometric progression

Suppose $$ax^2+bx+c$$ is a quadratic polynomial (where $a$, $b$ and $c$ are not equal to zero) that has real roots. Prove that $a$, $b$, and $c$ cannot be consecutive terms in a geometric sequence. ...
1
vote
2answers
26 views

Finding whether a piecewise function is even or odd

A periodic function with period $2\pi$ is defined by $f(x)=1$ in the interval $ a\lt x \lt b$ and $f(x)=0$ elsewhere. Can the function be even or odd? If not why not and if so, for what values of ...
1
vote
2answers
28 views

Function that fulfils the equation

Prove that for all $x\in\mathbb{R}$ there exists only one $y=y(x)$ that fulfils the equation: $$3x+e^x=y+e^y$$ I am completely lost with that. What should i do?
0
votes
0answers
10 views

Real-valued function with kind of additivity

I am looking for (family of) real-valued functions $f:\mathbb{N}^k \rightarrow \mathbb{R}$ such that for any $i$ one can compute $f(n_1, n_2, \dotsc,n_i+1, \dotsc, n_k)$ knowing only $f(n_1, n_2, ...
3
votes
2answers
60 views

How to find the range of a function $y=\frac{(x-1)(x+4)}{(x-2)(x-3)}$

I have the following function: \begin{equation} y=\frac{(x-1)(x+4)}{(x-2)(x-3)} \end{equation} It's easy to find it's domain, which is: $\mathbb{R} - \{3,2\}$. I know how to find the range of easier ...
2
votes
2answers
41 views

A circle of radius $r$ is dropped into the parabola $y=x^{2}$. Find the largest $r$ so the circle will touch the vertex.

If $r$ is too large, the circle will not fall to the bottom, if $r$ is sufficiently small, the circle will touch the parabola at its vertex $(0, 0)$. Find the largest value of $r$ s.t. the circle will ...
1
vote
2answers
48 views

Show that there is a number $x\in[\pi/2, \pi]$ such that $\tan(x) = -x$

Can I solve this by the intermediate value theorem?
0
votes
0answers
3 views

Show that a function is quasi-concave by finding a transformation of the function

Assume the function u:R2+ -> R given by u(x1, x2) = (x1+1)(x2+1) Show the u is quasi-concave by finding a transformation of u that is concave Show (without using Hessians) that u is not a concave ...
0
votes
0answers
12 views

Prove that $f(x,y,z)=\ln x-\alpha\ln y-(1-\alpha)\ln z$ is quasiconvex

I am trying to show that $f(x,y,z)=\ln x-\alpha\ln y-(1-\alpha)\ln z$ defined for $x,y,z\in\mathbb{R}^+$ and $\alpha\in(0,1)$ is quasiconvex. This is equivalent to showing that the set ...
0
votes
0answers
45 views

Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
0
votes
2answers
16 views

Notation of a boolean function

I'm studying Boolean algebra but I was confused as the notation of a Boolean function. When I write/denote a Boolean function that way, what does that mean? $$ f: \mathbb{Z}^2_2 ...
0
votes
1answer
31 views

Composition of functions, examples

Let \begin{align} f(x) &= -1, \\ g(x) &= 3x, \\ h(x) &= \begin{cases} 0, & x \text{ even} \\ 1, & x \text{ odd} \end {cases} \end{align} now how do you find: $$(f \circ g \circ ...
11
votes
1answer
105 views

If every point is a local maximum, is it a step function?

What are the functions $f:\mathbb R\to\mathbb R$ such that every point is a local maximum? Certainly, $f(x)=c$ works for every constant. So does $\lfloor x\rfloor$, as does ...
0
votes
0answers
33 views

MatLab Proper Way to Use Trapz with Angles

I have a discrete function, say $B(\phi, θ)$, which is a $361 \times 181$ (rows for $\phi$, columns for $\theta$) matrix. In fact, this function has more than two dependent variables but I want to ...
0
votes
0answers
35 views

For which natural numbers $m,n>1$ does the inequality $2\uparrow^m n>f_m(n)$ hold?

Denote $$f(n,m):=2\uparrow^{m-1} n$$ (See : Wiki ) and $$g(n,m):=f_m(n)$$ (See : Wiki ) It is straightforward to show $f(n,m)<g(n,m)$ for all $m,n>1$ via induction. But for which $m,n>1$ ...
0
votes
2answers
25 views

drawing square root of an unknown function with a double root.

Without the equation of the function $y=f(x)$ and only the diagram, how would you draw $y=\sqrt{f(x)}$ and why? I am most interested in root if it looks like a double root but not necessarily a ...
-2
votes
1answer
22 views

Greatest integer functions

If $A=\{1,2,3,4\}$, then which of the following are functions from $A$ to itself? I. $f_1=\{(x,y) \mid x+y=5\}$ II. $f_2=\{(x,y) \mid y<x\}$ I haven't got an idea of this question and I need ...
1
vote
2answers
61 views

In what sense could $x=y^5$ not be a function?

I encountered a test question that asked which of the following does not represent a function: $y=x+5$ $x = y + 5$ $y=5^x$ $y=x^5$ $x=y^5$ According to the answer, (5) does not. I am reasonably ...
1
vote
2answers
37 views

Is this function a bijection?

$f: \Bbb N \to P(\Bbb N)$ be given by $f(n) = \{n+1,n+2,n+3,\ldots\}.$ From general intuition and reasoning I think the function is not injective here is my work. If $n = 1$ $f(1) = ...
0
votes
0answers
33 views

What is this curve?

A quick question. I have the following plot: Just looking at the coloured band where it transitions from red to blue, it looks like a function I have seen somewhere before (rotated 90deg ...
0
votes
0answers
24 views

The function $g(x)$ is in $\Omega(x^c)$ as $x\rightarrow 0$ for all $c>0$. Does this imply that $\lim \inf_{x\rightarrow 0} g(x) > 0$?

Consider a function $g: \: \mathbb R^+ \mapsto \mathbb R^+ $. For any $c>0$ this function is in $\Omega(x^c)$ as $x\rightarrow 0$. That is, given $c>0$ there exist $L$ and $x_0$ such that $g(x) ...
1
vote
2answers
44 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
0
votes
1answer
40 views

Fenchel Duality in Prof. Bertsekas' lecture

Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed. For convenience, the relevant part is shown as: I am confused in two things: The ...
4
votes
3answers
241 views

How to prove this identity of ceiling function?

My book writes down this identity of least integer function: $$\lceil x\rceil +\left\lceil x + \frac{1}{n}\right \rceil + \left\lceil x + \frac{2}{n}\right \rceil + \cdots +\left\lceil x + \frac{n ...
46
votes
6answers
2k views

Proving that a function is odd

Assume that there exists a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective and satisfies $$ f(x) + f^{-1}(x)=x $$ for all $x$. Here $f^{-1}$ is the inverse function. Show that $f$ is odd. This ...