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

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2answers
50 views

Why should the solutions of $(\sin x)^2 = 0$ be rejected in the equation $((\sin x)^2)(\csc x + 1) = 0$?

Q: Determine the number of solutions for $((\sin x)^2)(\csc x + 1) = 0$ over the interval $0 \leq x < 2\pi$ with the correct reasoning. Correct answer: There is one solution because the solutions ...
-1
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0answers
53 views

help in proving converge of this sereis, please??? [closed]

I have $f(x)$. I know this $$|f'(x)|≤ C < 1 , 0≤C , x ∈R and I have this sequence $$a_1=x\\ a_2=f(x)\\ a_3=f(f(x)),\\\vdots\\ a_n=f(\cdots(f(x))\cdots)$$ I need to prove this sequence ...
2
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2answers
34 views

Solve the following (logarithmic) function for x

$x^{log_{2}x}+16x^{-log_{2}x} = 17$ Looks horrible, I started by removing the exponents: $e^{ln(x)*log_{2}x}+16e^{-ln(x)*log_{2}x}=17$ | ln() $ln(x)*log_{2}x-16ln(x)*log_{2}x=ln(17)$ $ln(x)*log_{...
0
votes
4answers
49 views

Small problem about domain of a function .

I want to know that whether $f:\mathbb{R}^2/\lbrace(0,0)\rbrace \to \mathbb{R}$ defined by $f(x,y) = \arctan(\frac{x}{y})$ is a function or not? I think this is very silly problem but i think it is ...
0
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0answers
36 views

Derivatives that are tangent to the original function

I was recently studying parabolas $ f(x) = ax^2 + bx + c $ whose derivative $f'(x) = 2ax + b$ is tangent to itself -- one example would be $f(x) = x^2 -6x +10;$ it is easy to see that if $c = a + \...
1
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1answer
43 views

Functions invariant under scaling

Which functions are invariant under the transformation $$f(x)=af(bx)$$ for constants $a$ and $b$? Are functions of the form $cx^n$ and $de^x$ the only analytic ones (as in having a power series ...
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3answers
62 views

Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
0
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1answer
35 views

Find the number of elements in range of $g(f(x))$

Let $f(x)$ and $g(x)$ be bijective functions where $f:(a,b,c,d)\rightarrow(1,2,3,4)$ and $g:(3,4,5,6)\rightarrow(w,x,y,z)$ respectively.Then,find the number of elements in range of g(f(x)). I have a ...
1
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1answer
31 views

For what integral value of $n$ is $3\pi$ the period of the function $\cos(nx)\sin(5x/n)$?

For what integral value of $n$ is $3\pi$ the period of the function $\cos(nx)\sin(5x/n)$ ? What should be the correct approach to this problem?Will taking the LCM of the periods of the two functions ...
2
votes
2answers
42 views

Solve the following (logarithmic) function for $x$

$(\log_{3}x)^{2} - 3\log_{3}x + 2 = 0$ We may not use many rules, so I would start by ignoring the ^(2), ignore -3* but take ...
0
votes
1answer
24 views

How do I find the period of the function $\tan{\pi/2[x]}$?

How do I find the period of the function? $$\tan{\frac{\pi}{2}[x]}$$ What are the factors that I must take care of? (Maybe its simple but i'm not getting it methodically.$2$ seems to work though) [] ...
0
votes
0answers
14 views

Finding and proving upper bound of specific function

Following function is given: $$ f : \mathbb{N{}} \rightarrow \mathbb{R^+} , n \mapsto \begin{cases} n! & \text{for } 1 \leq n \leq 17\\ 2^{2^n}& \text{for } 18 \leq n \leq 42 \\ \log_2 n &...
1
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0answers
38 views

How do I write a function that maps a variable to a set?

I have a function $\Gamma$ that maps elements from $N$ to a (possibly empty) subset of $N$. The number of elements in the resulting subset depends on which element of $N$ we are dealing with, i.e. $\...
2
votes
0answers
35 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
2
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1answer
51 views

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
0
votes
2answers
42 views

Proving $f(n)=100n+5 \neq \Omega(n^2)$

I have to prove that: $$f(n)=100n+5 \neq \Omega(n^2)$$ What I tried: let's assume that $f(n)=100(n)+5= \Omega(n^2)$. Thus, there must exist some positive constant $c$ and $n_0$ such that, $$0 \leq ...
0
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1answer
28 views

All linear functions are homogeneous of degree one?

I was looking through the Wikipedia page of "Homogeneous functions" and it stated that any linear function that maps V onto W is homogeneous of degree one. However, when I try to apply the definition ...
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2answers
56 views

When to rationalize to repair continuity, and why does it work?

I was working on a question out a GRE math prep book: "Find the inverse of $f(x) = \frac{x}{1-x^2}$ that works for all $x \in \mathbb{R}$ where $f$ is defined over $(-1,1)$" (works meaning is well ...
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3answers
46 views

How do I find Big O notation for this function?

How do I find Big O notation for this function? $$ n^4+100\cdot(n^2)+50 $$ In the book I am following, I got the following solution: $n^4+100(n^2)+50 \leq 2(n^4) \ \forall \ n \geq 11$ $n^4+100(n^2)+...
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1answer
25 views

Question about continuous onto maps of homeomorphic spaces.

If $f:(A,T) \rightarrow (B,T_1)$ is continuous and onto, and $$(A,T) \cong (C,T_2) \land (B,T_1) \cong (D, T_3)$$ $$\Rightarrow \exists g: (C,T_2) \rightarrow (D,T_3)$$ that is continuous and onto.
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1answer
36 views

Prove any function can be written as a composition between an injective and a surjective function.

Given an arbitrary function $f:A\rightarrow B$, write it as a composition between an injective and a surjective function, respectively.
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0answers
40 views

Prove that $\{f^{-1}(B_i)|i\in I\}$ is a partition of X.

Could someone confirm what I have shown is sufficient in proving what was asked? I have no other way of checking my proofs and any help would be appreciated. Thank you for your time. Let $f:X\...
0
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1answer
22 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
9
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4answers
167 views

Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$

Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions? $$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
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1answer
17 views

Function exercise check-up

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
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1answer
74 views

What's the order of operations when dealing with function composition?

Given $f:[0,1]\rightarrow \mathbb{R}$ and $g:[0,1]\rightarrow [0,1]$, $g(x)=x^2$. Which of the two equalities is true? 1)$f^2(x^2)=f^2(g(x))=(f^2\circ g)(x)$; 2)$f^2(x^2)=f(x^2)\cdot f(x^2)=f(g(x))...
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0answers
14 views

Notation for the index of minimum value of several variables

Assume we have several variables of the form $d_c$ which namely can be $d_1$, $d_2$, ..., $d_n$. I want to use mathematical notation to show for which index $c$ the value of $d_c$ is minimal for all ...
1
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1answer
39 views

Find function for graph

I would like to find a function for the following graph: I have drawn the graph myself, so not all subtle bends are to be replicated. I have noted the important points the graph should have in the ...
2
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3answers
72 views

How to find range of $\frac{\sqrt{1+2x^2}}{1+x^2}$?

How to find range of $$\frac{\sqrt{1+2x^2}}{1+x^2}$$ ? I tried put it equal to $y$ and squaring but I'm getting $4$th degree equation.
1
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4answers
61 views

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
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0answers
26 views

square integrable function?

If I want to find out for which $\alpha$ the function $f:B_1(0)\to\mathbb{R}$, $f(x)=x|x|^\alpha$ is in $L^2(B_1(0))$, where $B_1(0)\subseteq \mathbb{R}^n$, can I do something like this: $$\int_{B_1(...
0
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3answers
56 views

Inverse of $f(x) = 2x^2+8x+13?$

How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far: $y = 2x^2+8x+13$ $x = 2y^2+8y+13$ $x-13 = 2y^2+8y$ $x-13=y(y+8)$ This is where I got stuck. To be clear, I want ...
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0answers
17 views

How do I solve this equation $f(x, y) = x - y^3 + y$ local for $h(x)=y$?

How do I solve this equation $f(x, y) = x - y^3 + y$ local for $h(x)=y$? $y^3+y=x$ What next?
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1answer
33 views

Normal vector on a plot

Do a sketch of $f$ with the equation $f(x,y)=0$. Give in all non singular points of the curve a normal vector. $f(x,y)=x^{3}-x-y$ How can I do this thing with normal vector? I know that singular ...
3
votes
1answer
125 views

How many possible functions?

Take $f:\{1,2,3,4,5,6,7\}$ to $\{0,1,2,3,4\}$ How many such functions satisfy the cardinality of the pre-image of the set $\{3\}$ is equal to $3$. I thought it would be $35$, i.e :$7\choose{3}$ ...
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2answers
30 views

Marginal revenue of a monopolist [closed]

A monopolist faces a demand function $Q=4000(p+7)^{-2}$. If she charges a price of p, her marginal revenue will be: (a) $p/2+ 7$ (b) $2p+ 3.50$ (c) $p/2-7/2$ (d) $-2(p+7)^{-3}$ Correct answer is ...
1
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0answers
21 views

Calculate Density of Values in Cellular Automata

I am working with a special cellular automata that uses hexagonal cells rather than square cells, a hexagonal grid, rather than a square grid, and the set of complex numbers, rather than a finite set, ...
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2answers
34 views

Determine a and b so that function is continious

$$ g(t)= \begin{cases} 2t^2 ;& t<-1 \\ at ;&-1<t<1 \\ bt-\frac 12 ;&t>1 \end{cases} $$ How can I determine $a$ and $b$ so this function $g$ is continuous at whole $\mathbb R$. ...
1
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0answers
45 views

Sigmoid function with fixed bounds and variable steepness [partially solved]

(see edits below with attempts made in the meanwhile after posting the question) Problem I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a ...
10
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3answers
1k views

A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
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3answers
74 views

Domain of the function $\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$

What is the domain of $$\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$$ the answer is $(-\infty,\infty)$. Now the polynomial has degree $12$. Also it's continuously increasing from $1$. So I thought there ...
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1answer
29 views

Determining if a function is onto

If our range such as in the question below is all the real numbers excluding $0$, to determine if a function is onto we must ask if all real numbers excluding $0$ can be mapped to at least one value ...
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0answers
14 views

Resolution function explicity [closed]

Examine where the equation $f(x,y)=0$ locally by $y=h(x)$ can be resolved. Calculate in all these places $h'(x)$ by implicit differentiation. Enter the resolution function(s) $h(x)$ explicitly if this ...
0
votes
6answers
92 views

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$?

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$ ? All I could till now : LHS =$2\sin{x}+\cos{x}$ Since, $−\sqrt{5} \leq 2\sin{x}+\cos{x} \leq \sqrt{5}$ So a ...
2
votes
2answers
38 views

A question on mapping inside unit disc

For an analytic function $f:\bar D→\bar D$ where $\bar D$ is the closed unit disc centered at origin.Suppose $\bar D=D\cup$$\delta D$, where $\delta D$ is the boundary of open disc $D$ and $f$ is onto,...
2
votes
1answer
58 views

What are all the different classes of functions upon real numbers and what do they mean, exactly? [closed]

I have been hearing terms like "piecewise C1", "continuous", "linear", "piecewise constant", "trigonometric", "logarithmic", "exponential", "elementary", etc. functions for many years. I know what ...
4
votes
0answers
61 views

Prove that $f$ is invertible

Did I show enough to prove $f$ is invertible? Alternatively is there a more efficient way to do so? Thanks in advance for any help. Let $f : X \rightarrow Y $a nd $g : Y \rightarrow X$ be ...
2
votes
2answers
56 views

If $C\cap D=\emptyset$ Prove that $f^c(C)\cap f^c(D)=\emptyset$

Is this the proper way to go about proving this? By showing $C\cap D$=$f^c(C)\cap f^c(D)=\emptyset$? Any feedback would be greatly appreciated. I don't have any other way of getting feedback for my ...
3
votes
2answers
160 views

Is it ok to use Kronecker delta function to find if one of its variables belongs to a half open interval?

Kronecker "delta" function is generally defined as $\delta(i,j)=1$ if $i$ is equal to $ j$, otherwise $0$. How about if $j$ is not an integer? I mean let $j$ is a half open interval defined as $j=(0,...