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

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0
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1answer
48 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$?
2
votes
2answers
42 views

Example of limit of a function

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand example 4.2.2 on page 105. The aim of the example is to show that: $$ \lim_{x \to 2} g(x) = 4 $$ ...
1
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0answers
44 views

Solve the function from the composition [duplicate]

I have equations as follows $$f(f(x))=x^2+x$$ Then solve for $f(x)$. Can anyone give some hints about this question?
1
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1answer
37 views

Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
4
votes
1answer
60 views

Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$?

The exercise asks me this: Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$? ps: $f: \mathbb{R}\to \mathbb{R}$ I really don't know how to start :c, I appreciate hints.
5
votes
2answers
124 views

Inverse of $f(x)=x^n+x$ on $[0,\infty)$

Fix integer $n > 1$. The function $f_n(x) = x^n + x$ is monotone increasing on $[0,\infty)$, and so has an inverse $f_n^{-1}(x)$ that is also monotone increasing on $[0,\infty)$. I'm interested ...
2
votes
0answers
35 views

Description of a Space of Functions

Here is my question: Denote by $V$ the following space of functions on $\mathbb{R}$: $f\in V$ if and only if there exists a nonnegative integer $k$, complex numbers $a_1,\ldots,a_k$ and purely ...
2
votes
1answer
37 views

Are these functions identical?

Suppose we have an identity of the form $$x e^{f(x,y)}+y e^{g(x,y)} \equiv (x+y)e^{h(x,y)},$$ for all $x,y\in D$ where $D$ is some domain. Does this imply that $f(x,y)\equiv g(x,y)\equiv h(x,y)$ in ...
2
votes
1answer
18 views

Vertical asymptote (or any?) at removable discontinuity

If I have a removable discontinuity, do i have any kind of asymptote? I originally thought no, but this confused me a bit: http://www.purplemath.com/modules/asymtote4.htm Close to the bottom, it ...
14
votes
3answers
183 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
0
votes
4answers
54 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
18
votes
4answers
483 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
18 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 ...
3
votes
3answers
103 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
43 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
92 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?
1
vote
1answer
23 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
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0answers
17 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
11 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
20 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
74 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 ...
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1answer
17 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
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0answers
12 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
69 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
33 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
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1answer
38 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
27 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
42 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
263 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
57 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
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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
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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
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1answer
65 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
25 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
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0answers
10 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
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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
59 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
55 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 ...