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

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12 views

Question on Partial Derviatives

For function $f(x,y) = x^2 y$ The partial derivatives for $x$ is $2.x.y $. I'm new to such math equation and i'm learning them now. May i know why is it so? Thanks!
2
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2answers
25 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
0
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1answer
20 views

Find a function that maps x,y to $[0, n ( n + 1) / 2)$

Can you find me a bijective function that maps positive integers $x, y$ such that $0 \leq x < y \leq n$ to integers in $[0, n(n+1)/2)$ to use as a hash function?
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2answers
12 views

Prove if $g \circ f$ is $1-1$ and $f$ is onto, show that $g$ is $1-1$

Let $f: A \rightarrow B$ and $g: B \rightarrow C$. $g \circ f: A \rightarrow C$. But where do I use the fact that $f$ is onto?
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1answer
14 views

Domain of a multiple logarithmic function.

Find the domain of the following function: $f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$ My text provides a solution which goes like: => ...
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0answers
12 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
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2answers
16 views

functions in form of $f\circ g$

Express the function in the form $f\circ g$. (Use non-identity functions for $f$ and $g$.) $F(x)=(6x+x^2)^4$ I understand you have to find what each $f(x)$ equals and $g(x)$ equals but not really sure ...
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1answer
16 views

What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
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2answers
30 views

Hey guys. Given the graph below, find the equation of the transformed parent function. [on hold]

It would be great if there is a detailed explanation. Also, is there a standard method I can use to answer all kinds of graphs including exponents and logs? Thanks
2
votes
1answer
42 views

Is root of a function differentiable?

Let's assume a function $f(\alpha,\theta)$ always has a single zero wrt $\alpha$: $\forall \theta, \exists \hat\alpha_\theta$ such that $f(\hat\alpha_\theta,\theta)=0$. Let's now consider this root ...
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0answers
9 views

Seam Carving - Energy functions. How do they work?

I have been taking an interesting in dynamic programming and more specifically Seam Carving. For those who are not aware what this is, please look here and for some more detailed information here. If ...
1
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1answer
32 views

Can this function be a density function of a continuous random variable X?

F(x) = 0, if x < 1 F(x) = 1, if 1<=x<=2 F(x) = 0, if x>2 I think it could be, as long as the integral is 1. Any ideas?
2
votes
1answer
37 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
2
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0answers
35 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
1
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1answer
31 views

Showing the surjectivity of a function

$f:\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}$ is defined as $f(n)=(2n,n+3)$ $\mathbb{Z}$ means integers. I showed the injectivity but i'm confused with the surjectivity. Suppose that ...
0
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0answers
32 views

Difficult and strange functions question

I have an expression $$ x = \pm \frac{n}{2\pi}\cos^{-1}\left[\frac{\cos(2\pi y/n)+\cos(2\pi z/n) + sec(\pi/n)\cos(2\pi y/n)\cos(2\pi z/n) - 1}{1+sec(\pi/n)}\right]$$Where n, y and z are positive ...
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1answer
15 views

Function| Domain & Range

Suppose that A={0,1,2,3,4}, B={2,3,4,5}, f={(0,3),(1,3),(2,4),(3,2),(4,2)} Find the domain and range of f. f(1)=? f(2)=? My opinion is; f(1)=3 and f(2)=4 If I am right, what would be f(2) if ...
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1answer
17 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
5
votes
2answers
113 views

What is $f(f^{-1}(A))$?

Suppose that $f : E \rightarrow F$. What is $f(f^{-1}(A))$? Is it always $A$? $f^{-1}$ is the inverse function. This is not a homework, I'm confused by this statement.
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2answers
51 views

Proof of period of $f(ax+b)$

I have been taught that $f(x)$ is called a periodic function with period $T$ if $$f(x)=f(x+T)$$ This I understand completely. Also I have been taught that $$f(ax)=f(ax+T/|a|)$$ if $f(x)$ has a period ...
1
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2answers
26 views

The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$), let $\mathcal{F}$ be the set of ...
0
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1answer
9 views

Invertible or Non Invertible function?

If $f:$**R —> R** is defined by $f\left(x\right)=x^2+1$, then what are the values of $f^{-1}\left(17\right)\:$ and $f^{-1}\left(3\right)\:$ . My textbook arrives at the following answer: ...
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3answers
33 views

How do I find a function given the conditions as follows : [on hold]

Value of function on $(-\infty,k)$ is $0$ Value of function on $(l,\infty)$ is $1$ Function is of the form $e^{f(x)}$ and is continuous in the region between $l$ and $k$ as well.
4
votes
2answers
95 views

Proving that the cardinality of a set is even

Let $E$ be a set and $f:E\to E$ be a function such that $f\circ f=Id$. Let $A=\{x\in E, f(x)\neq x\}$. Suppose that $A$ is finite. Prove that the cardinality of $A$ is even. My ...
1
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1answer
43 views

Can injective function has an element that maps to nothing?

Can injective function has an element that maps to nothing? I don't think this violate the definition of injective function. If that is the case, is it possible for a function to be bijective but its ...
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0answers
19 views

A question on relating function on $N$-Sphere with a function on $\mathbb{R}^{N-1}$

I have a function $f:\mathbb{R}^N\to\mathbb{R}$, and at any given point $P\in \mathbb{R}^N$, I have a scalar quantity (a property of $f$) defined on all unit vectors $\hat{a}$ at $P$, and it can be ...
0
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2answers
17 views

which functions have unique fixed points on the stated intervals [on hold]

h(x)=x^3-1, x belongs to [1,2] I think this function has, but the answer says no. Any advice?
2
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1answer
50 views

$|\{0,1\}^\infty| = |\mathbb{R}|$?

Let $\{0,1\}^\infty$ = {$(a_n)_{n \in \mathbb{N}}; a_n \in \{0,1\} \forall n \in \mathbb{N}$} Is there a bijection between $\{0,1\}^\infty$ and $\mathbb{R}$? I thought about something like this: If ...
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2answers
23 views

Proof of Divergence Criterion for Functional Limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand corollary 4.2.5 on page 107. To be more specific, let me first write down the theorem that precedes ...
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3answers
17 views

Concurrency of lines

If the three lines: $$x\sin^2 \theta + y \cos^2 \theta = 1$$ $$x \cos^2 \theta + y \sin^2 \theta = 1$$ $$lx + my + n = 0$$ are concurrent then which of the following is true? a) $l+m=n$ b) ...
1
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1answer
33 views

Proof - Inverse of linear function is linear

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection ...
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0answers
28 views

How to find max and min bounds of a uncertain function

First I would like to say that I have searched the for uncertain fitting, robust fitting, linear optimization, convex optimization, etc. But I'm lacking the knowledge to solve this problem, and I need ...
4
votes
2answers
48 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
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0answers
25 views

Type of equation

I have an equation and I don't know what is the exact scheme of solving it? I mean "nonlinear" or "implicit"? $$ K([XI+(1-X)O ]^m)-K([XR+(1-X)P]^m)-\frac{\Delta t}{2}[(I+R)-(O+P)]=0 $$ $O$ is the ...
3
votes
8answers
198 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 $y$? The explicit question out of the ...
4
votes
2answers
59 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: $$ \lim_{x \to 2} g(x) = 4 $$ ...
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0answers
46 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
38 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
63 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
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2answers
131 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
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0answers
40 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
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1answer
42 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
19 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 ...
15
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3answers
203 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
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5answers
64 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) - ...
21
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4answers
695 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?
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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
104 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
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3answers
50 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
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2answers
93 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?