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

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3
votes
4answers
92 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
0
votes
0answers
33 views

Inclusion Mapping?

Is the hooked arrow map notation not supposed to mean an inclusion mapping? His definition is clearly showing inputs from $\mathbb{R}^2$ who's images are elements in $\mathbb{R}^{3}$ with fixed $z$ ...
1
vote
1answer
26 views

How do I “stretch” and “compress” a piecewise function?

I have Googled a few times and experimented on Desmos, but both attempts were to no avail, and now I come here. How is a piecewise function transformed to be "stretched" or "compressed"? What about ...
2
votes
0answers
38 views

How to apply the Identity Theorem to this function?

Given the function $f(z)=\exp\left(z^2-\cos\left(iz\right)-4\right)$ with the domain $|z|<10$, if we try to apply the Cauchy integral formula, we'll see that f(2) "will be" $$\frac{1}{2\pi i}\int_\...
0
votes
1answer
33 views

Composite functions yeah

I'm trying for the GRE so that I can apply for grad school in 2017. I've been working well through calculus and algebra. I'm making good progress but functions has been a challenge. Take this for ...
1
vote
0answers
17 views

Function output by parameter - value relations

I have a parameter set, P; p1, p2 ... pn And also I ...
0
votes
1answer
29 views

Minimum modulus principle - looks like a counterexample?

The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
0
votes
0answers
34 views

help on proving converging of sequence, please

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 ...
0
votes
0answers
27 views

Minimum norm of analytic function may not be achieved on the boundary of its domain

I need to show that the minimum modulus of an analytic function may not be achieved on the boundary of its domain. I'm stuck with this question, would appreciate if someone could help me with it. I ...
-4
votes
1answer
51 views

Piecewise Contextuous Functions - A possible new branch of functions?

Let us define f(x) as the following: $$f(x) = \begin{cases} g(x) & \text{if f(x) is being floored ($\lfloor x \rfloor$)} \\ h(x) & \text{if multiplication by anything other than 1 is being ...
1
vote
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 ...
2
votes
2answers
35 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
52 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
votes
0answers
37 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
vote
1answer
44 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 ...
0
votes
3answers
65 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
votes
1answer
37 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
vote
1answer
33 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
43 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
26 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
15 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
vote
0answers
39 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
38 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
votes
1answer
53 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
43 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
votes
1answer
31 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 ...
1
vote
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 ...
0
votes
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)+...
0
votes
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.
0
votes
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.
0
votes
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
votes
1answer
24 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 ...
10
votes
4answers
175 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? $$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
1
vote
1answer
18 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 ...
0
votes
1answer
77 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))...
0
votes
0answers
16 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
vote
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
votes
3answers
76 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
vote
4answers
65 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 ...
0
votes
3answers
63 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 ...
1
vote
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?
0
votes
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
128 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 :$7\choose{3}$ ...
-1
votes
2answers
33 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
vote
0answers
24 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, ...
0
votes
2answers
36 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
vote
0answers
50 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
votes
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 ...
-1
votes
3answers
76 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 ...
0
votes
1answer
30 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 ...