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

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16 views

How to solve asymptotic recurrence without using Master Theorem

I am working on the following problem. Consider the function $B:\mathbb{N}\to\mathbb{R}$ defined by: $$B(n) = \begin{cases} 1 & \text{if $n\leq 2$,}\\ 3\cdot B(\lceil n/\log_2 n\rceil) + n & ...
-2
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1answer
40 views

Functions - Algebra [on hold]

Two functions are defined by: $f(x) = 3x + 2$ $g(x) = x^2 - 4$ Find: (i) $fg(2)$ (ii) $gf(2)$ (iii) $fg(x)$ (iv) $gf(x)$ (v) the values of $x$ for which $fg(x)=17$
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2answers
43 views

Confusion with $O$ function

I read this identity in lecture notes and need help understand ing the $O$ function $$\sum_{1\leq d\leq x}\mu(d)\cdot \frac{1}{2}\left\lfloor\frac xd\right\rfloor\left(\left\lfloor\frac ...
1
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2answers
46 views

Show that $f: \mathbb N \to \mathbb N$, $f(x)=x^2$ is not onto

To begin, the definition of an onto (surjective) function is as follows. A function $\phi$ from $A$ to $B$ is surjective if for each for each $b$ in $B$, there exists at least one $a$ in $A$ such ...
2
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0answers
38 views

Conjecture of the general form of a power series

Relcently I met a power series(Source Link-Eq(4.1)) of the type $$ f(x)=1-x+\frac{1}{2}x^2+\frac{1}{4}x^3-\frac{1}{8}x^4-\frac{35}{128}x^5-\frac{157}{1024}x^6+\cdots $$ where $x$ is supposed to be a ...
-1
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0answers
8 views

How to write this z-transform function in the following form to find difference equation

I have the following transfer function: H(z) = (z^2+0.81)/(z^2 + 1) From what I have read in order to find the differnece I need to re-write H(z) in the following form: H(z) = (a0 + a1*z^-1 + ... + ...
2
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4answers
64 views

finding $\int {(2x + 5)^2}$

After slowly getting the hang of differentiation I have moved onto integration and I can't seem to understand this one. I know the answer is $$\frac{4x^3}{3} + 10x + 25x + C$$ I understand that ...
-5
votes
1answer
32 views

Given $f\colon S\to S$ is injective, is $f\circ f\circ f$ injective? [on hold]

Prove or disprove: For a mapping $f\colon S \to S$, if $f$ is injective, then $f \circ f \circ f$ is injective.
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1answer
16 views

How to prove a function derivability in a interval? [on hold]

How can I see if a function is derivable on a specific interval. What method do you use?
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3answers
22 views

Find the horizontal asymptote(s) of the function

I need to find the horizontal asymptote(s) of the following function: $$f(x) = \frac{-2e^x + x^2}{3e^x + 5}$$ This needs to be done using limits, and I know I need to apply the limit as x approaches ...
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0answers
37 views

Completely expressible functions

All we know about Sheffer's stroke and Peirce's arrow. Each of these functions allows us to express any other function in Z[2] ring. Also, next function in Z[3] has the same properties: ...
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1answer
16 views

$f(x)= \min\{ x-\left\lfloor x \right\rfloor ,-x-\left\lfloor -x \right\rfloor \} \quad $ for $-2 \le x \le2$

If $f(x)= \min\{ x-\left\lfloor x \right\rfloor ,-x-\left\lfloor -x \right\rfloor \} \quad $ for $-2 \le x \le2$ Then how to find number of solutions of the equation $x^2+[f(x)]^2$=$1$ in {$-1\le ...
0
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1answer
19 views

Composite function (State the domain)

The function $f$ and the composite function $g\circ f$ are defined by $f(x)=3x^2+2,x\in \mathbb R$ and $g\circ f(x)=9x^4+9x^2+2,x\in \mathbb R$ respectively. Find the function $g$ and state the ...
1
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0answers
34 views

How to prove no integer roots for this Polynomial? [duplicate]

Let $P(x)$ be a polynomial with integer coefficients. It is known that $P(a)=P(b)=P(c)=-1$ where $a,b,c$ are distinct integers. Prove that $P(x)$ does not have integer roots.
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1answer
35 views

Square root of an even polynomial is holomorphic

Given an even degree polynomial $p(x)$, all of whose roots satisfy $|z| < R$. Explain why there is a holomorphic (i.e. analytic) function $h(z)$ defined on the region $R < |z| < ∞$ which ...
1
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1answer
34 views

Calculating cardinality of the following sets

I want to calculate the cardinality of the various sets such as: The set of continuous functions from $\mathbb R$ to $\mathbb R$. The set of continuous functions from $\mathbb Q$ to $\mathbb Q$ The ...
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0answers
35 views

How to stretch a function along $y=x$ diagonal line?

How to stretch a function along $y=x$ diagonal line? For example, for function $y=\sinh^{-1}\left(\frac{x}{2}\right)$.
2
votes
0answers
17 views

Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
0
votes
2answers
26 views

Find a function $g(x)$ satisfying the above conditions.

Find a function $g(x)$ satisfying the above conditions:- a)domain is $(-∞,∞)$. b)range is $[-2,8]$. c)$g(x)$ has a period $π$. d)$g(2)$=3. ATTEMPT: Since the function is periodic with period $π$ ...
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0answers
15 views

Finding the value of $K$ by the replacement of a function

if $e^{f(x)}=\frac{10+x}{10-x}, x\in (-10,10)$ and $f(x)=kf\left(\frac{200x}{100+x^2} \right)$, then $k=$ (a) 0.5 (b)0.6 (c)0.7 (d) 0.8 Answer: (a) I tried by taking natural log both the ...
0
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2answers
35 views

How Would I Graph This Exponential Function?

How Would I Graph This Exponential Function? $f(x) = \frac{-3}{2^{(x+2)}} - 1$, How I do it: I know that it is basically $-3\times ({\frac{ 1}{2}})^{x+2} - 1$, which is then $-3 \times 2^{-(x+2)} - ...
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0answers
30 views

Proofs needed for certain results related to functional equations

Today our maths teacher told us the following results without stating the proofs: (These are all polynomial or exponential functions) 1) $f(x+y)=f(x)+f(y)$ then $f(x)=kx$ 2) $f(x+y)=f(x)f(y)$ then ...
0
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1answer
20 views

Does the function have horizontal or vertical asymptotes?

So I'm analyzing some functions here and I need to determine whether or not they have horizontal or vertical asymptotes. The equations are: $f(x)=260$ $g(x)=1+24(0.9)^x$ $h(x)=f(x)/g(x)$ Now ...
2
votes
3answers
27 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
10
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1answer
186 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
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2answers
12 views

Horizontal and Vertical Asymptotes of functions

So I'm completing a chart analyzing the different properties of three different functions: $f(x)=\log(x^2+6x+9), g(x)=\sqrt{x^2 -1}$ (sorry not sure how to do square roots on here), $h(x)=f(x)(g(x))$ ...
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0answers
21 views

Counting the number of elements $x$ between $p$ and $p^2$ where lpf$(x(x+2))=7$

Let $p > 7$ be any prime. Let $f_7(p)$ be a function that counts the number of elements $x$ where $p < x < p^2$ and lpf$(x(x+2))=7$ where lpf is the least prime factor. It has been ...
7
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2answers
134 views

Polynomial Functional Equation.

Let $f(x)$ be a one-one, polynomial function such that $f(x)f(y)+2=f(x)+f(y)+f(xy) \ \forall \ x,y \in \mathbb R - \{0\}$, $f(1) \neq 1$, $f'(1)=3$. Find $f(x)$. I tried to find the degree of ...
0
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2answers
30 views

What's the minimum value of the following function?

So, I need to figure out the minimum value of this function: \begin{equation*} y=x^2-2(m+1)x+2m(m+2). \end{equation*} I tried with the y-coordinate of the parabola's tip, but all I get is the ...
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2answers
29 views

How to show $x,y,z \in A$ - Functions, Combinatorics

If $A \subseteq \{1,2,3,4,5,6\}$, how to show that for every $A$ there are $x,y,z \in \{1,2,3,4,5,6\}$, where $x,y,z$ can also be the same or at least not different from each other, and the following ...
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1answer
25 views

Show that the function $g$ is bijective

Let $~f\colon X \!\to\! \{i \in \mathbb{N}:\! 1 \leq i \leq n \}$ be a bijective function and $~x$ be an element of $~X$. Now define the function $g\colon X-\{x\}\!\to\!\{i \in \mathbb{N}:\! 1 \leq i ...
0
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1answer
22 views

What is this octagon constant and how do I calculate it for other 8*N-gons?

I'm drawing a circle with triangles in OpenGL and I am no good at maths. I've tried a couple of ways, one including the simple ...
3
votes
3answers
42 views

Find the Range of the function $f(x) = |x-6|+x^2-1$

find the Range of $f(x) = |x-6|+x^2-1$ $$ f(x) = |x-6|+x^2-1 =\left\{ \begin{array}{c} x^2+x-7,& x>0 .....(b) \\ 5,& x=0 .....(a) \\ x^2-x+5,& x<0 ......(c) \end{array} ...
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1answer
44 views

Maximum Value of function f

How can I find the maximum value of the function \begin{equation*} f(x,y) = x^2 - y^2 + 2xy - 2x - 2y + 1 \end{equation*} where $x^2+y^2 \leq 2x$?
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3answers
36 views

Trouble understanding One-One and Onto function.

So I have a question like this: Let $g$ be a function $g : \mathbb{Z} → \mathbb{Z} \times \mathbb{Z}$ such that $g(n) = (2n, n + 3)$. And I want to find if this is onto and one-one. But I'm ...
0
votes
4answers
48 views

How to find : Range of $cos(cosx) $

How to find : Range of $\cos(\cos x) $ My approach : Since $-1 \leq \cos x \leq 1$ we get $\cos(-1) \leq \cos(\cos x) \leq \cos(1)$ Is it correct? please suggest. Can we use the above method ...
2
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2answers
35 views

Continuity and differentiability of the function $x|x|$

Let $f:\mathbb R \to \mathbb R$ defined by $f(x) = x|x|$, Is the function continous at all points? If it is, then is it differentiable at all points? Yes, the function is continuous everywhere but ...
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1answer
16 views

Local Extremes and Differentiable Functions

A local extreme value is found in the interior domain of a differentiable function. A claim is made that the curve must have a positive slope on one side of the extreme and a negative slope on the ...
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0answers
57 views

I need a function with Sine like movements that starts from 0 (on Y axis) and goes to 1 (on Y Axis) while only using Multiplication [closed]

Can anyone writes a function that produces the same Y s that you see below between $x = 0$ and $x = 5$ using multiplication and not addition ? (It doesn't need to be a prefect match! something similar ...
5
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3answers
32 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
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3answers
141 views

Endomorphic Function Definition

I need to confirm my thinking on endomorphic functions. Since an endomorphism is just a surjective morphism on an object to itself in a category, can I alter the usual definition of a surjective ...
0
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1answer
24 views

Is this a proof that recursive definition of functions indeed defines a function?

Someone asked me how you prove that defining a function recursively actually defines a function, and then I tried to rigorously prove it. Is it right? Let $\mathbb{N}=\{0,1,2,\dots\}$. For any ...
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1answer
59 views

Curious formula for minimum?

A few years ago I derived the following formula which I just came across in my notes: $$\min(x,y)=\log\left(\frac{e^x+e^y}{1+e^{|x-y|}}\right)=y+\log\left(\frac{1+e^{x-y}}{1+e^{|x-y|}}\right).$$ Has ...
4
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1answer
40 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
3
votes
1answer
67 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
0
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0answers
77 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
0
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1answer
87 views

Inverse of $x^x$ [duplicate]

Since $x^x$ grows very fast, its inverse should accordingly grow very slow, possibly slower than $\ln(\ln(x))$. I am troubled with finding such an inverse: I only get to the point: $\ln(x)x=\ln(y)$ ...
3
votes
3answers
36 views

$\sqrt{y}+\sqrt{x}=\sqrt{A}$ … prove that x-intercept + y-intercept of any tangent = constant [closed]

This is equation of a curve $\sqrt{y}+\sqrt{x}=\sqrt{A}$ $A$ is constant $T$ is a tangent of the curve from any point on it $B$ is y-intercept of $T$ $C$ is x-intercept of $T$ ...
0
votes
1answer
30 views

General expression that represents a combined period of 2 sine functions

How to find the general expression that represents the combined period of $y=5\sin(\pi x/6)$ and $y=3\sin(\pi x/4)$? what are the limitations of this model?
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1answer
20 views

Need help with a hyperbola/parabola equation solving for an assignment

Hi so i was given my Math C assignment today and the moment i looked at question 1 i knew i had no idea what to do. This is the graph i was given (http://imgur.com/nRXOlJy). I was asked to provide an ...