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

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8
votes
3answers
156 views

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $ Is there elementary function of $f(x)$ satisfy the equation?
1
vote
0answers
14 views

Function to generate a score out of 100% based on other parameters

I am attempting to score an outcome out of 100%, which will be an evaluation of "risk" level. The factors would be (for example): if number of users increases, risk level increases if secure ...
2
votes
3answers
89 views

Partial derivative function definition paradox

I've pondered this question over quite alot and haven't been able to find an answer anywhere. I'm going to ask this question from the standpoint of basic thermodynamics. Let's say I define ...
1
vote
3answers
40 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
votes
2answers
89 views

Why is the cube-root of $x$ 'odd'?

I am trying to understand why $\sqrt[3]{x}$ is an odd function; can anyone explain how I could come to this conclusion?
3
votes
1answer
43 views

If $f$ is injective and $g$ is surjective, is $g\circ f$ bijective?

Let $f:A\rightarrow B$ and $g:B\rightarrow C$. If $f$ is injective and $g$ is surjective, is $g\circ f$ bijective? I believe this is false, and have a counterexample. It was actually easier to ...
1
vote
1answer
12 views

Intersection points of a function and its inverse.

Why is it that when $f$ is an increasing function then the points of intersection of $f$ and $f^{-1}$ lie on the line $y=x$?
1
vote
0answers
15 views

prove the following equivalence $\ln(-\frac{l-x+\sqrt{r^2+(l-x)}}{l+x-\sqrt{r^2+(l+x)}})={}$ArcSinh$(\frac{l+x}{r})+{}$ArcSinh$(\frac{l-x}{r})$

Hi guys i'm trying to prove the following equivalence but i'm having some problems: $$\ln\left(-\frac{\ell-x+\sqrt{r^2+(\ell-x)^{2}}}{\ell+x-\sqrt{r^2+(\ell+x)^{2}}}\right) = \operatorname{ArcSinh} ...
0
votes
0answers
10 views

Completeness condition for periodic function

I know that for a real-valued function set $\{f_n(x)\}$, its completeness condition is $\Sigma_n f_n(x)=\delta(x-x')$. That is, this condition guarantees that a well-behaved function can be write as a ...
0
votes
1answer
19 views

prove that $g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$

$g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$ iff $$g(x_1+a,x_2+a,\dots,x_n+a)=g(x_1,x_2,\dots,x_n)\forall a \in \mathbb R$$ Trial: Only if part : Consider $g$ is a function of ...
1
vote
1answer
70 views

how to show a function is bijection

I have taken two numbers $p$ and $r$ where $p,r\in A = \{0,1,\ldots,4i + 1\}$ where $i\geq 1$ and $q\in B = \{0,1,\ldots,n-1\}$. Let $X$ contains all elements obtained by cartesian product of $A$ and ...
-1
votes
1answer
16 views

Straight lines - point of intersection

Question: Two rays in the first quadrant: $$x +y = |a|$$ $$ax - y = 1$$ intersect each other in the interval $a \in (a_0, \infty)$, the what is the value of $a_0$? I don't even understand where to ...
1
vote
1answer
45 views

$f\in C(\mathbb{R})$. What does it mean?

$f\in C(\mathbb{R})$. What does it mean? My guess is "Differentiable on $\mathbb{R}$" but I'm not sure.. Thanks.
0
votes
0answers
11 views

What is the nature of this one dimensional function?

Let $\mathcal{S}$ be a 2-D convex set whose elements can be represented as $(x,y)\in\mathcal{S}$. Let $p_L$ and $p_U$ be two real constants such that $p_L\leq p_U$. For $p\in[p_L,p_U]$, I define the ...
1
vote
1answer
34 views

Prove that $|f ''|\ge 4$

Let $f(x)\in C^2:[0,1]\rightarrow\mathbb{R}$ satisfy $f(0)=0,f(1)=1,f'(0)=f'(1)=0$, prove that: $$\max_{x\in[0,1]}|f''(x)|\ge4$$ By using Taylor series I can prove that ...
2
votes
1answer
68 views

List of functions $f(cx) = C\cdot f(x)$

I was looking for some complex functions f(x), which satisfies the condition: $$\exists (c, C) \in \Bbb C^2 \backslash\{(1,1)\}, \forall x \in \Bbb C, f(cx) = C\cdot f(x)$$ Till now I have got ...
0
votes
3answers
32 views

Compound functions: one to one and onto

Let $f: A \to B$ and $g: B\to C$ be maps. If $g(f(x))$ is one-to-one and $f$ is onto, show that $g$ is one-to-one I'm really not sure how to prove this. Would someone be able to walk me through ...
-1
votes
1answer
36 views

Determine if these correspondences on ${\mathbb Q}$ define functions

Given the correspondence $f: \Bbb Q \to \Bbb Q$, explain why $f$ is a function: a) $$f\left( \frac pq\right) = \frac {3p}{3q} $$ b) $$f\left( \frac pq\right) = \frac {3p^2}{7q^2}- \frac pq $$ ...
1
vote
1answer
24 views

exercise on pointwise convergence of an (easy) function.

Exercise 6.2.5. Taken from understanding analysis of Stephen Abbott For each n $\in N$, define $f_n on \ R$ by $$f_n(x) = \begin{cases} 1, & \mbox{if} \ |x| \ge 1/n \\ n|x|, & \mbox{if} \ ...
2
votes
1answer
41 views

Vertical asymptote, yes or no?

I am working on a problem that will highlight the importance of accuracy and the flaw in approximating certain numbers (very basic stuff). Say you have the following function $$f(x)=\frac{x^2 - ...
11
votes
3answers
262 views

What happens to a function when it is undefined?

If I have the function $$f(x) = {x^2 - 2 \over x + \sqrt 2}$$ this is undefined for $x = -\sqrt 2$, am I correct? Since the denominator would be zero. But the numerator is a difference of ...
-3
votes
2answers
54 views

Continuity of $f$ define by $f(x,y)=\frac{x^2+y^2}{\tan(xy)}$? [on hold]

consider the function $f$ define by $$f(x,y)=\begin{cases}\frac{x^2+y^2}{\tan(xy)}&\text{if}\ (x,y) \neq (0,0)\\ 0&\text{if}\ (x,y) = (0,0) \end{cases}$$ Prove that the function is ...
0
votes
0answers
22 views

Simple function notation

I'm just making my way in Math and I apologise for the ease of this question. I don't understand what $R^n$ in $f(x):R^n \rightarrow R$ actually means.
1
vote
6answers
264 views

Range of a trigonometric function

Question: Prove that: $$0 \leq \frac{1 + \cos\theta}{2 + \sin\theta}\leq \frac{4}{3}$$ I have absolutely no idea how to proceed in this question. Please help me!
0
votes
0answers
11 views

Is there a method to find the equation of a parabolic branch?

Does it exist a method to find the the equation of a parabolic branch? Assume we have $f(x)\rightarrow +\infty$ when $x\rightarrow +\infty$ and $\frac{f(x)}{x}\rightarrow +\infty$ when $x\rightarrow ...
0
votes
1answer
17 views

Find a function matching this scheme

As I'm building a model, I need a function equals to $+\infty$ in 0, $-\infty$ in R and 0 in $\frac{R}{2}$. So it will looks like this (sorry for this bad drawing I'm at work). It definitely looks ...
0
votes
1answer
14 views

K-Map reduction

There's an exercise which states that depending on certain rules a led(of different colour) shall turn on or not. There are four leds, so I've made four functions (One each led, through Karnaugh Map ...
2
votes
0answers
31 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
-1
votes
1answer
46 views

Suppose f(x) + 2f(1/x) = x . Evaluate f(5) in simplest form. [on hold]

If f(x) + 2(f(1/x)) = x, evaluate f(5). How can I go about solving this problem?
0
votes
1answer
64 views

Is $C(C(\mathbb R))$ notation for the set of continuous functions mapping $C(\mathbb R)$ to itself?

Given that in general functional analysis we have $C(\mathbb{R})$ being the set of all continuous functions, $f: \mathbb{R} \to \mathbb{R}$. However, could I use $C(C(\mathbb{R}))$ notationally to be ...
2
votes
2answers
51 views

Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...
0
votes
1answer
24 views

The meaning of product of functions in multivariable calculus

If $f$ and $g$ are $2$ functions $\mathbb{R}^n\rightarrow\mathbb{R}^m$ For $m=1$ and $n>1$ is $f\cdot g$ or $(f\cdot g)(x)$ defined for? Would that be a real number? And for $n=1$ and $m>1$, is ...
0
votes
1answer
31 views

does a positive/negative number cancel itself?

By positive negative I mean the function that looks like an addition sign $(+)$ with a subtraction sign $(-)$ right underneath it. does a positive/negative number cancel itself? As in $x$ ...
0
votes
0answers
9 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
1
vote
2answers
49 views

Successive Differentiation of $\mathrm{e}^{g(t)}$

I am trying to find the closed for solution for $A_n$. Assume $A_0 = g'(t)$, $A_1 = g'(t)$, and $$\dfrac{d^n}{dt^n}\left[e^{g(t)}\right] = A_n e^{g(t)}$$ The problem has a recursive relationship of ...
1
vote
1answer
30 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
1
vote
2answers
28 views

Prove $f$ isn't uniformly continuous

I already proved (followed by an hint) that $f(y)-f(x) > x(y-x)$ for all $y>x>0$. I need to prove $f$ isn't uniformly continuous on $(0, \infty)$. What I did: Lets assume by contradiction ...
4
votes
2answers
57 views

Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
1
vote
0answers
54 views

Is 1/x the “slowest” asymptotically falling off differentiable function?

As a physicist, I tend to think about $\sim 1/x$ as the "slowest" fall-off of a "reasonable" function. Let us state this formally: $${\rm lim}_{x \to \infty} f(x) = 0, f(x) \in Reas \implies \exists A ...
1
vote
1answer
46 views

Is there a uniformly continuous function such that $a_{n+1} = f(a_n)$?

Let $a_{n+1} = a_n - a_n^2$ and $a_1 = \frac{2}{3}$. I already proved that $a_n \to 0$ Now I was asked, is there a uniformly continuous function such that $a_{n+1} = f(a_n)$? All I can think of is ...
4
votes
3answers
133 views

How to find a function that is given by an equation

I have a Lebesgue integrable function $f : [0,1] \to [0,1]$ that solves the equation $$ 4 x f(x^2 ) = f(x) + f(1-x)$$ for all $x \in [0,1]$. Is it possible to give an analytic expression for $f$ ? ...
3
votes
2answers
89 views

Behaviour of the function $\ln(1+ x^2)$

Thus function has derivative equal to: $\frac{2x}{1+x^2}$. This indicates that it will flatten out while approaching infinity, ie, should have an asymptote. Yet, the function does not have any real ...
2
votes
1answer
94 views

How to solve $\tan x=x^2$ in radians?

How to solve $\tan x=x^2$ with $x \in [0, 2\pi]$? I try with trigonometry and many ways but the numerical solutions seems to be difficult.
1
vote
1answer
37 views

Statistics Probability Density Functions with Mutliple Features (Multivariate Normal Distribution)

I'm looking for a beginner-friendly explanation on how this Probability Density function works when dealing with mutliple features and what the variables and terms mean in detail. I'm seriously ...
2
votes
3answers
73 views

Is this proof correct? Injective function $ f: A \rightarrow B \iff $ function $ g: B \rightarrow A $ is surjective

I've begun a course in "Real Analysis" recently and I have this trivial exercise. Could someone check if my proof is correct? Proposition: There exists Injective function $ f: A \rightarrow B \iff $ ...
3
votes
1answer
32 views

Algebra equation with functions, constraints and a graph.

Consider the function $f:[1,3]\to\mathbf R$, $f(x)=-x^4+8x^3+ax^2+bx+d$, where $a$, $b$, $d$ are real constants. Find the values of $d$ for which $f$ has 3 stationary points between $x=1$ and $x=3$ ...
1
vote
1answer
39 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
0
votes
1answer
10 views

Terminology for a set of functions formed from a basic set of functions and all their compositions?

Let suppose I have a set $A$ and a set of functions $S$ from $A$ to itself. I can define a new set $S*$ that, intuitively, is the set of all functions formed by composing zero or more copies of ...
0
votes
1answer
44 views

Find the total number of real solutions

Let $$f(x)=9^x-5^x-4^x-2\sqrt{20^x}$$ How to find the number of real solutions of the equation $f(x)=0$? I tried this way: At $x\to-\infty$ value of $f(x)$ is $0$. At $x\to\infty$ we have ...
-1
votes
2answers
65 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$