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

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1
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2answers
35 views

Finding example of a special type of continuous differentiable function

Give example of a continuous function (if exists) $f : [a,b]\to \mathbb R$ differentiable in $(a,b)$ such that $f(a)f(b) \ne 0$ , the set $A:=${ $x \in (a,b) : f(x)=0$ } is infinite but not an ...
0
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1answer
16 views

How can I prove this is surjective?

Let $f: \Bbb{R}- \{2\} \to\Bbb{R} - \{5\}$ be defined by $$f(x) =\frac{5x + 1}{x-2}$$ My understanding of proving surjections is that you must show $f(x) = y$ i.e that all elements in the domain ...
-3
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0answers
20 views

Polynomial Functions/Remainder Theorem Challenge Problems

Please any help would be greatly appreciated! Refer to the photo for the question below. I know how to do it when given only one factor. How should I do it for this case? Thanks in advance for the ...
0
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0answers
9 views

(Probably simple) question about exchangeability of random variables

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
0
votes
2answers
55 views

Rookie looking for help with basic mathematical problem

This should probably be very simple, but I'm just not very skilled in math :S. I want a function that takes one variable, x, ranging from 0-1. As the input approaches 0 so should the output. As the ...
-4
votes
1answer
22 views

advanced algebra function word problem [duplicate]

can you please explain the answer and why you got it? the distance in feet $d(t)$ a dropped object falls in $t$ seconds is given by the function $d(t)=16t^2$. suppose you drop a ball from a height of ...
0
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1answer
24 views

prove or disprove the statements about functions [on hold]

Is there a non-zero function $f:R\rightarrow R$ such that $f(x+y)=f(x)+f(y)$ for all $x,y \in R$, $f(xy)=f(x)f(y)$ for all $x,y \in R$, $f(x+y)=f(x)f(y)$ for all $x,y \in R$, $f(xy)=f(x)+f(y)$ for ...
0
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1answer
16 views

Is there a way to do this? Fixed deduction for x rounds where total = fixed amount

I am trying to calculate the reduction amount / step per round for the given: rounds = 1000 points = 80 starting at reward = 1 point So from round 1 which has a reward of 1 point deduct a fixed ...
1
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3answers
62 views

Set notation and mappings question

Good evening. I have a question. Suppose I have two sets, $A=\{1,2,3,4\}$ and $B=\{5,6\}$. I want to write the notation for a function that takes each element in $A$ and assigns to it a value in $B$. ...
-3
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0answers
20 views

let c(x) be the cost to produce x tires, and let r(x) be the revenue in hundreds of dollars [on hold]

Let $c(x)$ be the cost to produce $x$ tires, and let $r(x)$ be the revenue in hundreds of dollars. $\displaystyle R(x)=\frac{-x^2}{2+5x}$ $\displaystyle C(x)=\frac{3}{2x+3}$
1
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2answers
31 views

Proving that: 800 + n log n + 200√ n log n = Θ(n log n)

I am trying to prove that 800 + n log n + 600sqrt(n)*log(n) = Θ(n*log n) (where log is base 2) Basically so far, I've reduced this expression into an ...
0
votes
1answer
31 views

Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R} $ with $x \mapsto x^2$ for all $x∈R$ find the ...
-1
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1answer
34 views

Could someone walk me through determining the domain of arcsin x-2/3? [on hold]

What will be the domain of this function? $$ \arcsin {x-2\over 3} $$
1
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2answers
17 views

stereographic projections find the function

The problem is in the image. I need help. I have no idea how to do this.
1
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1answer
25 views

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
1
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1answer
33 views

Does $f(2x) \in Θ(f(x))$ always hold?

If $f(x)$ is continuous and increasing positively, does $f(2x) \in Θ(f(x))$? I am convinced that this is false but I am stuck on the proof. $$0 \le c_1 f(x) \le f(2x) \le c_2 f(x)$$ $$0 \le c_1 \le ...
-2
votes
2answers
36 views

Area functions, Find a formula for A(x)

Let $A(x)$ be the area between the function and the $x$-axis and between the $y$-axis and the vertical line at a given $x$. Consider the following function. $$f(t) = \begin{cases} - 2t + 8, & ...
1
vote
2answers
31 views

If f(n) = O(n), does log(f(n)) = O(log n)?

I have been trying to find a counter-example to prove this is false, however I feel that I am going in the wrong direction. f(n) = O(n), does lg(f(n)) = O(lg n) ...
0
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1answer
20 views

Prove that X = Y

Let $X=\{n^3 + 3n^2 + 3n \;|\, n \geq 0\}$ and $Y=\{n^3 - 1 \;|\, n > 0\}$. Prove that $X=Y$. I solve this question by putting $n=0$ in $X$ So, $x= 0+0+0 = 0$ and I put $n=1$ in $Y$ and so: ...
1
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2answers
24 views

I need to find the formula for h(x) for all x

we're given a function $h(x)=\begin{cases}1/2&\text{for}&0\le x<2\\0 &\text{otherwise}\end{cases}$. Then we are told to define the function $\displaystyle g(x)= ...
1
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1answer
29 views

List of well-known submodular function in physics, statistics, math?

Can you please share a list of well-known submodular functions (have the diminishing return property) that you know? In physics, stats, math, etc? I am searching for a submodular function for my ...
0
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2answers
18 views

Finding the equation of oblique asymptote of non-rational function

I have the function: $f(x)=2x-2^{x}+2$ I know that this function has an oblique asymptote, but all the tutorials I can find on google, are with rational functions with the form: ...
2
votes
0answers
44 views

If the set of values , for which a function has positive derivative , is dense then is the function increasing ?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ?
0
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1answer
11 views

On the existence of a non-constant sequence whose differentiable image converges [duplicate]

Let $f: [a,b] \to \mathbb R$ be a function differentiable in $(a,b)$ , then is it true that there is a non-constant sequence $(x_n)$ in $(a,b)$ such that the sequence $\big(f(x_n)\big)$ is ...
0
votes
2answers
37 views

Getting inverse of polynoms with trigonometric functions

I'm trying to get the inverse of $$f(x) = \cos(x) + 3x$$ I tried it by definition of $\cos(x)$ with no luck: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+...$$
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votes
1answer
51 views

Maths as an engineering of functions [on hold]

Sometimes I was thinking why there is not a bigger perspective of maths as a engineering instead of a collection of tools or a way to investigate deeper relations between components. I dont know any ...
2
votes
3answers
49 views

In what sense is a function on a circle the same as a $2 \pi$ periodic function on $\mathbb{R}$?

I was reading the appendix of Elias M Stein's Fourier Analysis and before proving the approximation lemma the author mentions the following Recall that a function on a circle is the same as a $2 ...
0
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0answers
20 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
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2answers
25 views

there is not such function on real numbers

Prove that there isn't a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f>0$, $f'>0$ and $f''<0$. Any suggestion will be appreciated.
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1answer
15 views

Remainder of Polynomials

A polynomial $P(x)$ of degree $n \geq 2$ has a remainder of $9$ when it is divided by $(x+2)$ and a remainder of $-1$ when it is divided by $(x-3)$. Find the remainder of $P(x)$ when it is divided by ...
1
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1answer
24 views

Determining functions types. one - one , onto and bijective

Could anyone give an idea to start up with this question: Let $f$ be a function $f: \mathbb{R}^3 \to \mathbb{R}$ such that $f(x,y,z) = xyz$ . How to verify whether it's one to one or onto function?
0
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1answer
19 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
0
votes
1answer
38 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
1answer
18 views

What is a polynomially bounded function?

I know this question has been answered before, but I didn't understand the answers and my reputation is too low to comment, since I'm new to stack exchange. Polynomially bounded (I'm pretty sure) ...
0
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3answers
44 views

proving that if $f(x)$ odd then $f(0)=0$

Need to prove that if $f(x)$ is an odd function that defined in the point: $x=0$, So $f(0)=0$. I know that odd function is: $f(-x)=-f(x)$ And that $f(x)=0$ is an odd function but dont know how to ...
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votes
2answers
20 views

Find injection, surjective, and bijective functions from S to T and vice versa [on hold]

Let S = {1, 2, 3, 4, 5} and T = {A, B} I have to find how many functions there are in total and also how many injection, surjection, and bijection functions there are from S to T and also from T to ...
1
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4answers
23 views

Range of inverse trigonometric function

Find the range of $y$. $$y=\tan^{-1}\left(\frac{2x}{1+x^2}\right)$$ I used the following approach: Let $$x=\tan\theta$$ $$\therefore \theta=\tan^{-1}x$$ Since the principal solution of $\tan^{-1}$ ...
0
votes
1answer
16 views

show a cartesian product in function is injective or surjective?

I had previously figured out injectivity/surjectivity on basic functions but I am stumpted when it comes to showing functions which are cartesian products are injective/surjective. The first one: ...
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votes
1answer
28 views

trigonometric function and identities [on hold]

If sin(x) = 1/3 and sec(y) = 17/15, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.) sin(x − y) I have absolute no idea what this ...
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vote
2answers
38 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
0
votes
1answer
19 views

Quadratic Functions: Determine the value of b

I'm having trouble with this question and I'm not sure what to do. Would appreciate any one who helps me out. Question: The point $(-2,1)$ is on the graph of the quadratic function: $f(x) = -x^2 + bx ...
1
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0answers
29 views

A function of class $C^2$

Problem: given a function $F:\mathbb{R}^2\mapsto\mathbb{R}$ of class $C^2$, with $F(0,0)=0,\nabla F=(2,3)$, shown that a surface $F(x+2y+3z-1,x^3+y^2-z^2)=0$ can be given locally at $(-2,3,-1)$ as ...
1
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0answers
17 views

If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)$, then the curve $P(x,y)=\alpha$ cuts $y=f^{-1}x$ at?

If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)\forall x,y\in\mathbb R$, then the curve $$y^2+\int_0^x(\sin t+a^2t^3+bt)dt=\alpha,\alpha\in\mathbb R^+$$ cuts $y=f^{-1}x$ ...
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0answers
25 views

For a boat to float in a tidal wave the water must be 2.5 meters deep… (trig questions)

$$y=5+4.6\sin\left(\frac{t}{2}\right)$$ What is the period in hours? Simply $p=2\frac{\pi}{n}=4\pi$ which is $\pi$ per hour If the boat leaves the bay at midday what is the latest time it can return ...
2
votes
1answer
61 views

If $f \circ g$ is surjective, $g$ is surjective

If $f \circ g$ is surjective, $f$ is surjective. If $f \circ g$ is surjective, $g$ is surjective. $\textbf{Part 1:}$ Let $f:B \to A$ and $g:C \to B$. Assume $f \circ g$ is surjective. Since ...
-3
votes
1answer
65 views

If fg is surjective, then g is surjective.

Either prove or give a counterexample to the "converses" of exercise 2 on page 17 If fg is injective, f is injective. If fg is injective, g is injective. If fg is surjective, f is surjective. If fg ...
2
votes
1answer
33 views

How many of these are surjective?

Let $A=\{a,b,c,d\}$ and $B=\{e,f,g\}$. How many maps are there from A to B? How many of these maps are surjective? $\textbf{Part 1:}$ There are 4 elements in A and 3 elements in B. Thus there are ...
0
votes
1answer
25 views

Determining if a function decreases exponentially

Define a function: $f(x) = \sqrt{\frac{e^{-kx}}{1-e^{-kx}}}$ where $k > 0$. Does this function decrease exponentially? EDIT: Sorry, I meant to ask just if it decreases exponentially.
-3
votes
1answer
28 views

Calculus net signed area [on hold]

so this homework problem I have asks me to find the net signed area. So, what I did was 1+2+3+4+5+6-5-4-3-2-1 = 6, but that is wrong... Why is it wrong? Thanks!
1
vote
2answers
43 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...