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

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0
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0answers
19 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
0
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0answers
5 views

How to translate a function along its normal in any given point?

Let f be some differentiable function in R. Let n(x) be the normal of the tangent at (x, f(x)). How can you translate each point of f along the normal at this point given a certain distance?
0
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1answer
38 views

How do you evaluate this function? [on hold]

Given the function $g(x) = x^2 + 2x$, evaluate: $\displaystyle\frac{g(x)-g(a)}{x-a}$, where $x\ne a$ This is how far I got: $\displaystyle\frac{x^2 + 2x - a ^ 2 - 2a}{x-a}$, where $x\ne a$
6
votes
2answers
72 views

Increasing function with $f'(x)=f(f(x))$

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
1
vote
1answer
29 views

Is this non-constant function periodic for every definable number?

Given the set $\mathbb{D}$ which contains all definable real numbers. The definition must not be infinite long. E.g. it contains $12$, $-3$, $\frac{1}{12}$, $\sqrt{2}$, $\pi^2$, $i+e$, Chaitin's ...
1
vote
1answer
38 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 ...
1
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2answers
20 views

Do both sets have the same cardinality?

Well I was trying to find out whether the two sets $[n, n+1]$ and $[n, n+1]\cup \{n+2\}$ has the same cardinality. If you add another infinite set(not any random one) to $[n, n+1]$, for example $[n, n+...
0
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5answers
59 views

What is the domain and range of $y = \sqrt{9 − x^2}$?

What is the domain and range of real function $f(x) = \sqrt{9 − x^2}$? In order to find the function's domain, you need to take into account the fact that, for real numbers, you can only take the ...
2
votes
3answers
53 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 ...
1
vote
2answers
63 views

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
0
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0answers
27 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
25 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 ...
-4
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0answers
27 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 something other than f(x) is being added to f(x)} \\ h(x) & \text{if multiplication is being applied to f(x)} \\ j(...
0
votes
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 ...
0
votes
1answer
32 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 ...
2
votes
0answers
36 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_\...
1
vote
2answers
49 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
13 views
0
votes
1answer
28 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
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0answers
30 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 ...
-1
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0answers
53 views

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

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
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0answers
22 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 ...
7
votes
3answers
102 views

Is there an expression for a function that maps integers to one and non-integers to zero?

Is there a function that can be built with addition, multiplication, exponentiation, trigonometric functions, integrals, (and all of their inverses i.e subtraction, division, taking logarithms, $\...
1
vote
2answers
75 views

reflecting a function about a line

Say you have a function $f(x)$ and a line $g(x)=ax+b$. How do you reflect $f$ about $g$? I am apparently supposed to write more text, but the line above is all I am after, hence I wrote this sentence ...
0
votes
1answer
429 views

excel- function with multiple variables

So I'm not good with excel (computers in general) and can do some things but this one is out of my league. This is the problem: The cost of a used car is highly correlated with the following ...
3
votes
3answers
2k 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 about ...
0
votes
4answers
46 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 ...
2
votes
2answers
32 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
3answers
61 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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0answers
35 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
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
1answer
162 views

What should be number of integral values of n?

If the period of the function $\cos(nx)\sin(5x/n)$ is $3\pi$ then what should be number of integral values of $n$ ? My approach : I tried like period of $\cos(nx)$ is $2\pi$/n and $\sin(5x/n)$ is $2\...
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) [] ...
38
votes
8answers
2k views

Does this pattern have anything to do with derivatives?

In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern ...
0
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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 &...
2
votes
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 $(Ax)t = t^2x(a)$, ...
1
vote
0answers
37 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
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0answers
34 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 ...
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0answers
95 views
+100

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
0
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2answers
40 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 ...
1
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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 ...
0
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1answer
35 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
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2answers
1k views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ^{x-1}$...
9
votes
2answers
193 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
2
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1answer
35 views

Finding the equation of a polynomial

A quadratic function with a minimum of 5 has zeros at -4 and 2, find the equation of this function. This is impossible, correct?
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 ...