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

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1answer
35 views

Solve for $x$ in $(3sf)$, where $\cos(x) - \tan(x) = 3 $.

The problem I am struggling with to solve is this. I have already tried to solve it however I end up with quadratic $\sin$ or $\tan$ whichever way you do it, which does not help. Solve for $x$ ...
2
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1answer
54 views

Show a function is one-to-one and onto

Consider f: ℝ{1} → ℝ{1} given by f(x) = x/(x-1) Show that f(x) is one-to-one and onto. What I have: If a function is one-to-one then it follows that if f(a) = f(b) then a=b. If a function is onto ...
2
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2answers
30 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
0
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1answer
60 views

Prove that the function is constant $f(x)=\arcsin2x\sqrt{1-x^{2}} - 2\arcsin x$ [closed]

Prove that this function is constant: $$f(x)=\arcsin\left(2x\sqrt{1-x^{2}}\right) - 2\arcsin x$$
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2answers
82 views

$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded? [duplicate]

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded? I think that it will always ...
1
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1answer
31 views

Prove Bijection in roots of unity function

Given $k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} $. Probe that if $n$ and $m$ are coprime, the function $f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta$ is bijective. ...
0
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1answer
38 views

Boolean function on $\{0,1\}^n$ comprising just binary AND and OR gates

Let $f:\{0,1\}^n\to\{0,1\}$ be a boolean function computed by logical circuit comprising just binary AND and binary OR gates (assume that the circuit doesn't have any feedback). Let ...
5
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1answer
67 views

function inequality $f(x+y)+y \leq f(f(f(x)))$

$f(x+y)+y \leq f(f(f(x)))$ find all possible solution for $ f: \mathbb {R} \rightarrow \mathbb {R}$
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2answers
360 views

How to show this function is an injection (one to one)?

Consider the function $f: \mathbb N$ × $\mathbb N$ → $\mathbb R$, $f(a,b) = a+b \sqrt{11}$ How do I show this function is an injection (one to one)?
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1answer
24 views

Extending $\mathbb R$ for the benefit of the unitary pulse function

The unitary pulse function (or sample function) is defined as follow: Let's $\newcommand\R{\mathbb R}d_1:\R\to\R$ be a positive intrgrable function such that $$\int_{-\infty}^{\infty}d_1(x)dx=1.$$ ...
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2answers
52 views

If $ 2<x^2<3$ then find then no of solutions which satisfy that $({x^2})=1/(x)$

If $$ 2<x^2<3$$ then find then no of solutions which satisfy that $$({x^2})=1/(x)$$ where (x) stands for fractional part.! I tried to convert it into greatest integer but i got no ...
0
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1answer
58 views

Prove that [x/y] is a primitive recursive function

Prove that [x/y] is a primitive recursive function using this theorem: If $g(x_1,...,x_n)$ is primitive recursive, then $f(x_1,...,x_n)=\sum^{x_n}_{i=0}g(x_1,...,x_{n-1},i)$ is also a primitive ...
1
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0answers
114 views

composite functions and injection

I want to know if my attempts at parts b,c are correct for the following problem. a)Prove that the composition of one to one functions is a one to one function. b)Show by example that the converse is ...
0
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1answer
81 views

computability and uncomputability

1) Suppose $f$ is an increasing function from $\mathbb N \to \mathbb N$ $(i.e., if x\ge y, then \space f(x) \ge f(y)).$ Is there necessarily a program which computes $f$? 2) Suppose $f$ is a ...
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2answers
56 views

How to proof the randomness of a number sequence?

I've got a sequence of numbers generator by a "random number generator". Is there a way or a method to proof the randomness of the generator? How would I even compare randomness of generators? Or ...
6
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2answers
109 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
0
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4answers
60 views

Connected Subsets of X x Y

Let $X$ be a connected topological space and $f : X\to Y$ a map. Show that the graph $G(f)$, defined by $G(f) = \{(x; f(x)) \in X \times Y | x \in X\}$ is a connected subset of $X \times Y$. I ...
2
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1answer
56 views

Does there exist a suitable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n^m) = mf(n)$? I think no.

Any ideas how to prove that no injection $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is closed under multiplication by elements of $\mathbb{N}$ satisfies the following identity? $$f(n^m) = ...
1
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1answer
25 views

Number of distinct functions on a vector space $\mathbb{N}^3$

Let $k$ be an integer at least $4$ and let $[k] = \{1,2,\ldots,k\}$. Let $f:[k]^4 \to\{0,1\}$ be defined as follows: $$f(y_1,y_2,y_3,y_4) = 1\ \mathrm{iff\ the\ y_i's\ are\ all\ distinct}$$Now for ...
0
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2answers
38 views

Function that is onto and continous

Can there be a continuous onto function f : R -> R \ Q? I know if a function existed it would map reals to the irrationals. Any ideas?
2
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2answers
104 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...
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2answers
27 views

Multiple group homomorphisms

I need to prove or disprove the following: Let f and g be group homomorphisms from G to H where (G, ·)(H, ∗). Define h : G → H by h(x)=f(x)∗g(x). Then h is also a homomorphism. I want to say that h ...
0
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1answer
36 views

What is the rule for multiplying in integrals?

What is the rule for finding the integral of the product of two functions? Like this: ∫f(x)g(x)dx
2
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5answers
62 views

What functions $f: A \to B$ and $g: B \to A$, satisfy a restriction such that $f$ is not invertible but $f \circ g=id_B$?

I am caught up on the notation of $id_B$. I'm thinking that $f=x^2$, or something along those lines, but not so sure as to what $g$ may be.
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1answer
34 views

Finding a y(x) that satisfies $ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $

I'm having problem with finding a y(x) that satisfies $$ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $$ Here is what I tried to do. $$ y(x) = \int_ \! \left(\frac{x} ...
1
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1answer
23 views

Wat is the correct term for 'historic average'?

I've got the following array of numbers: {2, 4, 8, 1, 3} I've got a function that will shows the 'historic average' for every number: ...
5
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2answers
93 views

Inequality associated with Fourier transform

Suppose $$\int_{I_1} x^2|f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|f(x)|^2dx$$ and $$\int_{I_2} x^2|\hat f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|\hat f(x)|^2dx$$ for interval $I_1, I_2$ centered at origin and ...
0
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1answer
34 views

Range of $f\colon\{2,4,6\}\to \Bbb N, x\mapsto 3x-4$

What is the range of the function $f(x)=3x-4$, when the domain is $\{2,4,6\}$? I have tried plugging in each intercept to get the answer, but I cannot figure it out.
1
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2answers
266 views

Injective and surjective functions on a matrix

Suppose we have a function $G:M_2(\mathbb R) \to S_2(\mathbb R)$ where $S_2(\mathbb R)$ is a symmetric matrix such that $ S_2(\mathbb R) = \left\{A = \begin{bmatrix} a & b\\ c & ...
1
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2answers
32 views

Expectation of function of two random variables

I have the random variables X and Y, with joint density function $f(x,y)$ over the plane $-\infty < x < \infty$ and $-\infty < y < \infty$. I am trying to find the expectation of ...
17
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1answer
230 views

Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
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2answers
28 views

Function and its inverse in Discrete Mathematics

Prove that : if $L \subseteq M,$ then $f^{-1}(L) \subseteq f^{-1}(M)$.
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3answers
44 views

second derivative of exponential $e^{x^2}+3x-2$

I have to find the first and second derivative of $e^{x^2}+3x-2$, the first one i can do ok but can someone please help me with the second. thanks
1
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1answer
102 views

What is $(g \circ f)(3)$?

If $f(x) = x^2$ and $g(x) = \sqrt{x+4}$ What is $g \circ f (3)$ ? $g(f(x))$ would be $(\sqrt{x^2+4})(3)$ correct? What would be my next move in figuring this out?
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1answer
37 views

Measureability of Simple Functions over different sets.

Suppose E and F are subsets of R (we do not know yet whether E and F are Lebesgue measurable). Suppose we also know the function $s(x) = 5\chi_{E} (x) + 2\chi_{F} (x) is Lebesgue measurable. Prove ...
0
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1answer
89 views

Isomorphism of Complete Graphs

I am struggling to understand the concept of isomorphism. By definition, if G and H are two simple graphs so that V(g) and V(h) are the number of nodes in G and H respectively, then isomorphism is ...
2
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1answer
55 views

Formal proof: $f: A \rightarrow B$ is injective if $f$ is surjective and $|A| = |B| <\infty$

Formal proof: $f: A \rightarrow B$ is injective if $f$ is surjective and $|A| = |B| < \infty$ How does one proof formally that $f$ must be injective if the above criterion is satisfied ? ...
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0answers
21 views

How can I find the probability of a value occuring in an interval of a given step function?

Knowing a step function $N(t)$ defined over the interval $[0,T]$, which takes values in $0, 1, ..., k$, how can I define the probability $p(n)$ for $0 \leq n \leq k$, which is the probability of a ...
2
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1answer
88 views

What's the minimum value?

What's the maximum and the minimum value of $x$? $$\frac{(\sqrt{100-x^2}+\sqrt{99+x^2})}{40} = \cos \frac{\pi}{x^2-2|x|+4}$$ I've done all what I could do but I failed. Any ideas? Thanks
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2answers
30 views

What's the differential of this function?

So, for some reason my business management major includes calculus and I am so not good at this, I have no idea what a differential is for, only that it is the variation between $f(x)$ and $f(x+h)$, ...
0
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1answer
135 views

Order preserving bijection between the integers and the rationals

Is there a bijecttion $f:\mathbb{Z}\rightarrow\mathbb{Q}$ s.t $f(x) < f(y)$ iff $x < y$
0
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2answers
126 views

Bijection from $f:\mathbb Z \to \mathbb N \times \mathbb N$

I'm struggling to solve the task of showing a bijective $f:\mathbb Z \to \mathbb N \times \mathbb N$... Have you any ideas? We currently discus the topic relation/function and inverse. In this ...
0
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1answer
26 views

Does existence of derivatives imply continuity? In regard to usage of Bernstein theorem for approximation using polynomials.

I have a function $f$ defined on interval $[a,b]$ and I know that all of it's derivatives exist and $|f^{(k)}(x)|>0$ for all $x \in [a,b]$ and $k \in \mathbb{N}$. Does this imply that $f$ is ...
0
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2answers
32 views

Absolute difference for proving continuity

A function $|f|:D\rightarrow\mathbb{R}$ is defined by $|f|(x)=|f(x)|$ for $x$ in $D$. Prove it is continuous if $f:D\rightarrow\mathbb{R}$ is continuous. In my professor's solution to this proof he ...
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0answers
30 views

Definition of Discontinuities and example

So definition of discontinuity of some function at some point goes like this: A function $f$ is not continious at point $a$, where $a$ is an element of $D(f)$, if there exists $x$, which is an element ...
1
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1answer
111 views

A polynomial is called a Fermat's polynomial…

A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0) = ...
1
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2answers
72 views

Proof for Image of Indexed Collection of Sets?

Trying to prove that if f is one-to-one, then $$f\left(\bigcap\{U_\alpha:\alpha \in \Lambda\}\right)=\bigcap\{f(U_\alpha):\alpha\in\Lambda\}$$ I am able to prove that: ...
0
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1answer
239 views

Functions (Onto and one to one) and Floor functions

Suppose you have: $$f:\Bbb R \rightarrow\Bbb Z \text{ where } f(x)= \lceil 2x-1 \rceil$$ Well, I know that this is not a one to one function, but I don't know how to show it's onto. Thea reason ...
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2answers
166 views

Which function is an injection but NOT A SURJECTION

Which function is an injection but NOT A SURJECTION? (1) $h:\mathbb{N} \rightarrow\ \mathbb{Z}$ $h(x) = x^2 + 5$ (2) $p:[0,\infty) \rightarrow\ [5,\infty)$ $p(x) = x^2 + 5$ I think (1) is ...
-1
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1answer
108 views

For a price and cost function, at what rate do the weekly sales change per week?

For a price function of $p=4000-25x$ and a cost function of $C=1800x+4500$, if the profit is increasing at a rate of $3000$ per week and the weekly sales are $x=32$ units, at what rate do the weekly ...