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

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1
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1answer
20 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
1
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2answers
41 views

Solve $1/x^2 = \sum_{i=1}^{n} \frac{1}{x+a_i}$ over $x>0$

Does the equation $\frac{1}{x^2} = \sum_{i=1}^{n} \frac{1}{x+a_i}, \qquad a_i > 0, \quad i=1, \ldots, n$ always admits one and only one solution $x^* > 0$? If yes, what is the most elegant ...
2
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3answers
35 views

limit of sin function as it approches $\pi$

In my assignment I have to find the Classification of discontinuities of the following function: $$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$ I wanted to look what happens with the value $x=\pi$ because ...
0
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2answers
19 views

Proper use of indicator function

Given a set $X$ and a subset $A \subseteq X$ the indicator function $\boldsymbol{1}_{A} : X \rightarrow \{0,1\}$ of $A$ is defined as $$\boldsymbol{1}_{A}(x) = \begin{cases} 1 & \text{if } x \in A ...
2
votes
1answer
22 views

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?

Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$? By the way, I do not know how the adverb "explicitly" is strictly ...
0
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1answer
17 views

find the Classification of discontinuities of a function

In my assignment I have to find the Classification of discontinuities of the following function: $$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$ I wanted to start with the value $x=0$ because the function ...
0
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2answers
17 views

tangent for 3-dimensional function?

How can I calculate a tangent at a point $(x_0, y_0)$ in the direction $(r_1, r_2)$ for a $3-$dimensional function $f(x,y)$? I thought: \begin{equation*} T: (x_0, y_0, f(x_0,y_0)) + k \cdot (r_1, ...
2
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0answers
26 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
-1
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1answer
33 views

How to prove this has no real solution?

How to prove that $\left\lfloor x \right\rfloor +\left\lfloor 2x \right\rfloor +\left\lfloor 4x \right\rfloor +\left\lfloor 8x \right\rfloor +\left\lfloor 16x \right\rfloor +\left\lfloor 32x ...
1
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2answers
30 views

Function Composition Thinking Problem

Here is the question: A banquet hall charges $\$975$ to rent a room, plus $\$39.95$ per person. Next month they will offer a $20\%$ discount off the total bill. Determine two equations, one for ...
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1answer
24 views

Find the centroid of the region under the graph of the function $ w(x) = 4.5 + a x^{3} $ between $ x = 0 $ and $ x = 5 $. [on hold]

I need to find the centroid to determine where the equivalent force is acting on the region under the graph of $ w $ between $ x = 0 $ and $ x = 5 $. The given information is $$ w(0) = 4.5 ~ ...
0
votes
2answers
31 views

Finding two functions $f(x)$ and $g(x)$

I am not sure how to approach this question. It asks to find $f(x)$ and $g(x)$ such that $h(x)=f(g(x))$, for each function: a) $$h(x)=\sqrt{x^2 + 6}$$ b)$$h(x)=\frac{1}{x^3}-7x+2$$ If someone ...
0
votes
1answer
21 views

Determine the value of combined functions with square roots

The question I have is to determine the value of $f(g(x))$ given $f(x)=\sqrt{16-x^2}$ and $g(x)=x^2$ I know generally how to tackle these kinds of questions, but I am not sure what to do when there ...
2
votes
5answers
94 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
0
votes
3answers
24 views

Find the domain of combined functions

I have a question asking to find the domain of $g(f(x))$ given $f(x)=2x^2+x$, and $g(x)=x^2+1$. I can easily do these questions in reverse when you have to find $f(g(x))$, but when having to find ...
2
votes
1answer
31 views

Open-Set Correspondence $\implies$ Continuity

I would like to show the following implication. Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}^m$. If $f^{-1}(U)\subset\mathbb{R}^n$ is open for every open $U\subset\mathbb{R}^m$, then ...
1
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1answer
12 views

Is it overkill to define the closure of a set $A,A\subseteq B$ by the union of the range of the recursive function $h(0)=A, h(n^+) = h(n)\cup f[h(n)]$

$f:B\to B,A\subseteq B$. Is it overkill to define the closure of a set $A,A\subseteq B$ over $f$ by the union of the range of the recursive function $h(0)=A, h(n^+) = h(n)\cup f[h(n)]$? I ...
0
votes
2answers
38 views

$A \subset B \implies f^{-1}(A) \subset f^{-1}(B)$

Prove: $A \subset B \implies f^{-1}(A) \subset f^{-1}(B)$ I am busy setting up a proof for Real Analysis, and have come to a point where I need to use the above statement. Intuitively, I ...
0
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1answer
38 views

What exactly is the distance of two elements in $C[0,1]$?

If $C[0,1]$ — the set of all continuous functions from $[0,1] \rightarrow \mathbb R$ — is equipped with the metric $||\cdot||_1$ (1-Norm), then what is the distance between ...
1
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1answer
9 views

Sequence of functions that extends the algebraic properties of exponents to higher level operators.

I was thinking about some simple algebraic exponent properties such as the following $$ z^{x+y} = z^xz^y $$ and I started wondering about analytically continuing this identity to "higher-level ...
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2answers
45 views

How many workers each company have? [on hold]

If total there are 90 workers between 2 companies and one company have 16 more workers then the other. How many each company have?
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0answers
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
votes
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 $fg(x)=17$
0
votes
2answers
41 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
45 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
votes
0answers
36 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 ...
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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
votes
4answers
62 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
31 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.
0
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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 ...
1
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0answers
29 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: ...
0
votes
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
votes
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
33 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
31 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
vote
1answer
33 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 ...
0
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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
16 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 $π$ ...
0
votes
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
votes
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)} - ...
0
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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
votes
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
184 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 ...
1
vote
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), $h(x)=f(x)(g(x))$ ...
0
votes
0answers
20 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
votes
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
votes
2answers
28 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 ...