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

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1
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3answers
47 views

how can I find the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , $ x \in (-1,1)$ [on hold]

I want to check the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , for $ x \in (-1,1)$ but i don't know what to do. Every theory I know it is not working. Can someone ...
2
votes
1answer
35 views

Find an injective function that maps $\mathbb{R} \to (-\infty, 0]$

I'm looking for any ideas as to a function which maps $\mathbb{R} \to (-\infty, 0]$. I considered $-|x|$ but realised that is not injective.
4
votes
1answer
112 views

A continuous onto function from $[0,1)$ to $(-1,1)$

How I can construct a continuous onto function from $[0,1)$ to $(-1,1)$ ? I know that such a function exists and also I have a function $\displaystyle f(x)=x^2\sin\frac{1}{1-x}$ which is ...
0
votes
1answer
39 views

Clarification of the topology lemma “Any continuous and open injection of the open disk extends over the circle”

My elementary topology 1 class last semester used the book "Topology: Point-Set and Geometric" by Paul Schick, and covered through the end of chapter 8. I am working through the rest of the book on ...
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0answers
21 views

what is Step function?A general definition of what it is. .And explain about it please. [on hold]

what is Step function?A general definition of what it is. .And explain about it please.
1
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3answers
134 views

Derivative of the following function (similar to Softmax)

I am having a hell of time trying to differentiate the following function with respect to x. Do you have any suggestions $f(x) = \frac{ w(i)^x}{ \sum\limits_{j} w(j)^x }$ where $w$ is a vector ...
0
votes
1answer
21 views

Derivatives relation

Please, help me solve the following problem: Suppose $$\frac{df(x,y)}{dx}>0, \qquad \frac{df(x,y)}{dy}<0, \qquad \frac{d^2f(x,y)}{dxdy}<0$$ Is it true that if $y_{2}>y_{1}>0$ then ...
2
votes
3answers
17k views

How many one to one and onto functions are there between two finite sets?

Suppose $X$ has $N$ elements and $Y$ has $M$ elements. How many one to one function are there from $X$ to $Y$? How many onto function are there from $X$ to $Y$? The number of one to one functions ...
0
votes
1answer
9 views

What if the input of a simple function question is X?

I know how to answer function questions when they are like: fg(3) when f(x) = x + 3 and g(x) = x^2 But what do I do when the question is like: fg(x) when f(x) = x + 3 and g(x) = x^2 Or for a ...
0
votes
1answer
423 views

How to solve for maximum area of a rectangle under a curve?

Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is $8^{-\frac{x}{5}}$. I would greatly appreciate a full explanation. I already ...
3
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4answers
66 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ ...
0
votes
1answer
21 views

Determine a curves position over another curve

if the curve of $y= mx^2 -2mx +m$ is over rhe curve of $y=2x^2 -3$, then the limits of the interval must be my attempt: I dont know which concept i have to use. I only know that discriminant is use ...
0
votes
1answer
13 views

Finding the upper tight bound of a mathematical function. (Big O)

I am trying to understand Big-$O$ notation through a book I have and it is covering Big-$O$ by using functions although I am a bit confused. The book says that $O(g(n))$ where $ g(n)$ is the upper ...
3
votes
4answers
74 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
1
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2answers
25 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
5
votes
2answers
394 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
1
vote
1answer
55 views

Engineering/mathmatics question

I have an equation $M(x)= -15.328x^2+176.44x-352.88$ (a parabola) and also $V(x) = -30.657x + 176.44$. I want to know how to find $x$ where the values of $M$ and $V$ combined are the lowest, I'm ...
1
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0answers
21 views

Point which do not lie in domain of $f (\frac{2}{x-2})$

If $$f(x)=\frac{1}{x^2-17x+66}$$ then the points which are not in the domain of $f (\frac{2}{x-2})$ are: $(A) \frac{7}{3}$ $(B) \frac{24}{11}$ $(C) \frac{8}{3}$ $(D) 2$ I have already found that ...
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1answer
19 views

Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$

Let $f:X \to Y$ be a map with $A_1,A_2 \subset X$ and $B_1,B_2 \subset Y$. Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$ where $f^{-1}(B) = \{x \in X: f(x) \in B\}$. Attempt: ...
0
votes
2answers
116 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...
2
votes
2answers
72 views

How many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. A few related questions inside.

I am trying to calculate how many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. I would like to see the explicit mappings and learn how you ...
-1
votes
0answers
24 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
1
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0answers
45 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
-1
votes
3answers
36 views

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$. [on hold]

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$.
0
votes
0answers
6 views

Find the maximum value of $f(x,y,z)$ on the interval $x_0<x<g^x(p)$, $y_0<y<g^y(p)$, $0<z<g^z(p)$, $p=p(x,y,z)$

First of all, sorry if I am misusing terms or any tags in the post; I am a bit out of my depths here so I'm just trying to explain things in layman's terms. Now, here's the problem: I am working on ...
0
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0answers
9 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
0
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0answers
26 views

How to prove a function is not positive definite [on hold]

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
1
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2answers
31 views

Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto

Let $f:A \to B$ and $g:B \to C$ be maps. Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto. Attempt: If $g \circ f$ is onto, then for all $y \in A$, $\exists x$ such ...
0
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1answer
33 views

Hi I was wondering if there is any algebraic way to find the zeroes of a cos/sin formula without using the unit circle? [on hold]

I understand how to find the zeroes using the unit circle or just graphing it for any matter for my equation. I was just wondering if there is a formula or an algebraic way I could find them. ...
4
votes
1answer
17 views

Constructing bijection from set of equivalence classes to another set

Suppose $f:A \to B$ is surjective. Define a relation on $A$ by setting $x\sim y$ if $f(x) = f(y)$. It is clear that $\sim$ is an equivalence relation on $A$. Let $\mathcal{E}$ be the set of ...
1
vote
1answer
158 views

How can functions disagree with the values of its expansions at some points on an algebraic curve

I found a curve, in which some function has at least two expressions, which differ infinitely much!! Is there any error in the thoughts? The curve is defined by "\begin{equation} ...
1
vote
3answers
220 views

Range of a complicated function

Is there any way to figure out the range of values of the function $$y=\frac{2}{x}\cdot \sin(x)?$$ The domain is so easy to know. It's all real numbers except $0$. However the challenging part is to ...
1
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2answers
39 views

If $ Q(x)= x^2-5x+1 $ , find $ \frac {Q(5+h)-Q(5)}{h} $

If $ Q(x)= x^2-5x+1 $ , find $ \dfrac {Q(5+h)-Q(5)}{h} $ can someone show the steps to reach the answer of $h+5$? I got it down to $\frac{h^2+5h-2}{h}$
2
votes
2answers
19 views

Function that returns 1 when a non whole number, 0 when whole number

The title in this case should be self explanatory. When $x$ has a fractional part greater than $0, y$ should be equal to $1$, and when $x$ is a whole number, $y$ is equal to $0$. Anything that gives ...
2
votes
2answers
10 views

Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$. Define $j: \bigcup\limits_{k \in K} A_k \to ...
2
votes
5answers
250 views

A-noncompact, Does there **always** exist a continuous function $f: A \to \mathbb R$ which is bounded but does not assume extreme values?

It's well known that if $ A \subset \mathbb R$ is compact then every continuous function $f:A \to \mathbb R$ is bounded and assume extreme values .So the obvious question is: Given any non compact ...
1
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2answers
41 views

Check if $f(x)=2 [x]+\cos x$ is many-one and into or not?

If $f(x)=2 [x]+\cos x$ Then $f:R \to R$ is: $(A)$ One-One and onto $(B)$ One-One and into $(C)$ Many-One and into $(D)$ Many-One and onto $[ .]$ represent floor function (also known as greatest ...
2
votes
2answers
2k views

Trying to figure out a formula with given input and outputs.

I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers. ...
1
vote
2answers
46 views

Showing there is a constant for which an inequality holds true

I'm supposed to show that for $x>0$ and $p>0$ there is a constant $C$ such that $e^x\ge Cx^p$. The constant $C$ depends on $p$ but not on $x$. After analysing the behaviour of the graphs of ...
1
vote
3answers
73 views

Evaluating the inverse trigonometric limit $\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}}$

$$ \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$ I was doing some questions on limits, I saw one in which there is $\arccos x$. I am stuck ...
0
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2answers
23 views

Number of solutions using graph

We have to find the number of solutions of $e^((-x^(2))/2)$ + $-x^2 =0$ I tried it and got one solutions by drawing graph. Is I have done correct ? My try is on :
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0answers
15 views

function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
0
votes
3answers
54 views

If $f(x+1)= -f(x-1)$, prove that the function f is a periodic function and its period is $4$

I sure do know that I should arrive to the point where $f(x)= f(x+T)$. I've tried replacing $x+1=a$ but the problem seems to get more and more complicated.
0
votes
2answers
39 views

Determining exact value of $\cos (A+B)$ in a specific quadrant

The question reads: Angles $A$ and $B$ are obtuse angles in quadrant 2 (II). If $\csc A = 3$ and $\tan B$ = -1/3, determine the exact value of $\cos (A+B)$. How would I take on this question? ...
0
votes
2answers
45 views

Domain of Integral $\int_{5}^{x} \frac {dt}{(1-t^2)}$

A function reads $$ F(x) = \int_{5}^{x} \frac {dt}{(1-t^2)} $$ Barrons says that the domain of F must be that $x >1$. But why can't $x$ be less than $1$ as well? As long as $x$ does not equal ...
1
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0answers
27 views

Discontinous and differentiable

If we have a two functions $f:[-1/2, 2] \to \mathbb{R}$ $g: [-1/2,2] \to \mathbb{R}$ $f(x) = [x^2-3]$ where $[ \cdot ]$ denotes greatest integer $g(x) = |x| f(x) + |4x-7| f(x)$ Now we have to ...
1
vote
3answers
28 views

State the domain of $f^{-1}$

$$f(x)=\sqrt{2x+5}$$ $$x \geq -2.5$$ State the domain of $f^{-1}$ \begin{align} \ x & = \sqrt{2y+5} \\ \ \Rightarrow f^{-1}(x) & = \frac{x^2-5}{2} \\ \end{align} The Mark Scheme says that ...
3
votes
3answers
53 views

Finding domain of $f \circ g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then $f ...
0
votes
2answers
23 views

An exhaustion of $C_b(\Omega)$

Consider the space $\Omega=\mathbb{R}^{\mathbb{N}}$ and the space $C_b(\Omega)$ consisting of all bounded continuous functions defined in $\Omega$. Actually we are considering in $C_b(\Omega)$ the ...
4
votes
3answers
130 views

If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$

Let $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ and it is given that $f(0)=1$ and $f'(0)=-1$, where $f'$ denotes first derivative. Find the value of $f(2)$ Could someone tell me how to use $f'(0)=-1$ ...