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Questions tagged [functions]

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

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7 views

How to calculate percentage distribution after changing the set

My first language is not English so I appologize if I use incorrect terminology. For the sake of argument: I have a table of 10 whole numbers 1..10. I need to assign each row a value ...
2
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2answers
20 views

Find the domain of the following function

The given function is: $f(x)=\sqrt{\log_{|x|-1}(x^2 + 4x +4)}$ My approach: The argument $x^2+4x+4>0$ for all $x\neq-2$ Also, the base $|x|-1$ should be greater than 0 and not equal to 1. $\...
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4answers
23 views

How to find the image of this function?

I have been struggling with this math problem for a few days. I really need some help. Find the image of $h : (0, 1) \rightarrow \mathbb{R}$ defined by $h(x) = 1/(x^2 + 8x)$ for $0 < x < 1$. ...
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1answer
17 views

Boolean function in place of Bernoulli Random variable

A Bernoulli random variable $X: \Omega \to \{0,1\}$ is a function that maps each element of sample space $\Omega$ to 1 with probability $p$ and to 0 with probability $(1-p)$. Suppose we have a ...
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1answer
22 views

Find $c$ and $n$ such that $\frac{x^3 \arctan x}{x^4 + \cos x +3} \sim cx^n$ as $x \to 0$

Where is my mistake in the below: $$\frac{1}{c} \lim_{x \to 0} \frac{x^3 \arctan x}{x^{4+n} + x^n \cos x + 3x^n} = \frac{1}{c} \lim_{x \to 0} \frac{x^4}{x^{4+n}+x^n \cos x + 3x^n} \\ =\frac{1}{c} \...
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1answer
38 views

About the funtions that satisfies $f(ab)\geq b f(a) + a f(b)$

I am doing some research in information theory related to the $f$-divergences and some of their properties. So we have a convex function $f:(0,\infty)\rightarrow \mathbb R$ such that $f(1)=0$, and ...
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0answers
14 views

Solving stereographic projection for central latitude $\phi_1$ and central longitude $\lambda_0$

For a given longitude and latitude and their projected point I want to know the central longitude and latitude of the stereographic projection. I used the formulars from wolfram alpha: http://...
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0answers
38 views

Two functions on $\mathbb N$ such that $a(f(x))=x$ and $a'(f(x))=x$?

Given function $f:\mathbb N→\mathbb N:x→x^2$, how does one find two functions $a:\mathbb N→\mathbb N$ and $a':\mathbb N→\mathbb N$, such that $a(f(x))=x$ and $a'(f(x))=x$? N being the set of natural ...
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1answer
25 views

name for a function that composes additively

Suppose that the function $T: \mathcal{X} \times \mathbb{R}^k \to \mathcal{X}$ has the property that $T(T(x, z_1), z_2) = T(x, z_1 + z_2)$. For example, rotations and translations have this property -...
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0answers
12 views

Show that a function cannot be expressed as a strictly increasing transformation of another function

Consider a function $\Phi: \mathcal{I}\subseteq \mathbb{R}\rightarrow \mathbb{R}$. Suppose (a) $\Phi(0)=0$ (b) $\Phi(1)=1$ with $0\in \mathcal{I}$ and $1\in \mathcal{I}$. Consider a function $\...
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2answers
20 views

Is there a name for function where the order of its arguments doesn't change its output?

Is there a name for a function $f$ such that $f(x_1, x_2, \ldots, x_n) = y$ for all permutations of $x_1, x_2, \ldots, x_n$? Or a better notation to express this property? If I called it a ...
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2answers
39 views

For which $a$ is this function increasing? $ f(x) = \left( \frac {a-2}{a-4}\right) ^{-x} $

For which $a$ is this function increasing? $$ f(x) = \left( \frac {a-2}{a-4}\right) ^{-x} $$ So first I would rewrite this as: $$ f(x) = \left( \frac {a-4}{a-2}\right) ^{x} $$ I was thinking ...
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1answer
44 views

Divide and conquer: Why $F(0) = 0$? [on hold]

Reading algorithms. Fibonacci example. I saw an example where it said $F(1) = 1$ and $F(0) = 0$. (Fibonacci) Why $F(1) = 1$ and $F(0) = 0$? From where does it originate? Thanks.
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3answers
61 views

Find the coefficient of $x^{10}$

We have been given the following function. $f(x)$= $x$ +$x^2$ + $x^4$ + $x^8$ + $x^{16}$ + $x^{32}$ + ...upto infinite terms The question is as follows: What is the coefficient of $x^{10}$ in $f(...
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0answers
35 views

Can someone explain how to go about solving this question? [on hold]

Let $m,n\in\mathbb N$. Define a mapping $f:\mathbb Z/n\mathbb Z\to\mathbb Z/m\mathbb Z$ by $f([a]_n)=[a]_m$. (a) Prove that if $m|n$, then $f$ is a well-defined function. (That is, prove that if $[...
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1answer
33 views

help with functions? [on hold]

Let $f: A \rightarrow B$ be a function and let $A_1,A_2 ⊆ A$. Prove that $$f(A_1 \cup A_2) = f(A_1) \cup f(A_2)$$
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1answer
37 views

Is $\frac{\partial^2_xf \partial_yf}{\partial_xf}$ a total derivative?

Say $f$ is a function of two variables $x,y$. Is $\frac{\partial^2_xf \partial_yf}{\partial_xf}$ a total derivative? Or can we show $\int dx dy\frac{\partial^2_xf \partial_yf}{\partial_xf}$ vanishes?
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1answer
53 views

how to prove a function is injective iff it is surjective [on hold]

Let X and Y be nonempty sets and suppose f: X→Y is a function. Define a new function F: P(Y)→ P(X) by F(B)=$f^{-1}$(B). Prove that F is injective if and only if f is surjective. Note: P(Y) and P(...
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1answer
12 views

How do I find the equation for any given function?

If I am given a graph with labeled points, and I’m asked to find the equation that represents the given function, how can I do so? Just by looking at points it would be nearly impossible, and trying ...
3
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1answer
37 views

Showing a group does not have a faithful finite-dimensional representation

I am having some difficulty with this problem. We are asked to show that there does not exist any monomorphism to GL(n, $\mathbb{C}$). The group in the domain is defined as follows: Let G = $(P(\...
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3answers
47 views

Evaluate $\lim_{x \to 4} \frac{x^4-4^x}{x-4}$, where is my mistake?

Once again, I am not interested in the answer. But rather, where is/are my mistake(s)? Perhaps the solution route is hopeless: Question is: evaluate $\lim_{x \to 4} \frac{x^4 -4^x}{x-4}$. My ...
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2answers
46 views

The inverse function of $f(x)=\sqrt{x^2+1}+2x$

Find the inverse function of $f(x)=\sqrt{x^2+1}+2x$ with $x \in \mathbb{R}^+$.
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2answers
19 views

Directional derivative and composite functions

We have a function $f(p_0)$ with $p_0 \in \mathbb{R}^n$, and the vector $\vec{v}$ with $||\vec{v}|| = 1$. The derivative of $f(p)$ to $\lambda$ in the point $p = p_0 + \lambda \vec{v}$, calculated for ...
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2answers
74 views

Just find number of its roots: $\sin{x}-x\cos{x}=0$

There is an other equation here: $$\sin{x}-x\cos{x}=0$$ Range for x : $[0,\frac{3\pi}{2}]$ Now we want to write the equation $f(x)$ like $h(x)=g(x)$ : $$\sin{x}=x\cos{x}$$ It is in a way that we know ...
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0answers
36 views

Give example of The function is continuous and its absolute value is constant and is f not constant? [on hold]

Give example of The function is continuous and its absolute value is constant and is f not constant ? In complex C I think it is sin x It is true or not?
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1answer
35 views

the difference of functions tends to zero

If for any $\epsilon >0$,there exists $a_0$(depends on $\epsilon$) such that $f(a_0)>1-\epsilon$,where $f$ is a real function.We also know that $f(a_0)\leq 1$,can we deduce that $f(a_0)^4=f(a_0)...
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1answer
53 views

Simplify $x^n + x^{n-1} + … + x^1 + 1$ [duplicate]

Can I somehow simplify $x^{n-1}+x^{n-2} + ... + x^{2} + x^{1} + 1$? I would like to have an explicit formula for that sum, but could not figure out a way to do so. Could you help me? Thanks!
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0answers
29 views

Generalizable combination of combinations ratio equation

I have been trying to solve the following problem for a few days and any help or direction towards combinatorics literature would be appreciated. Given a starting sequence of length n, an index ...
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2answers
32 views

How to show the image of $h : (0, 1) \rightarrow \mathbb{R}$ defined by $h(x) = 1/(x^2 + 8x)$ is equal to $(1/9, \infty)?$

How to show the image of $h : (0, 1) \rightarrow \mathbb{R}$ defined by $h(x) = 1/(x^2 + 8x)$ is equal to $(1/9, \infty)?$ Hi, I want to show that the image of $h : (0, 1) \rightarrow \mathbb{R}$ ...
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0answers
10 views

How to isolate y in tuppers self referential formula?

How would we go about isolating $y$ in the following inequality? $${1\over2} = \mathrm{floor}(\mathrm{mod}(\mathrm{floor}(y/17)*2^{(-17*\mathrm{floor}(x)-\mathrm{mod}(\mathrm{floor}(y), 17))},2))$$ ...
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2answers
27 views

Find $a,b,c$ if $f(4)=1, f(-2)=14, f(5)=-2$ and $f(x)=ax^2 + bx + c$

Would anyone please explain to me the steps needed in order to find $a,b,c$? I understand I must use elimination, but the answers I keep on getting are very unrealistic. Thanks in advance.
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0answers
34 views

Plot and “meaning” of the function $ f(t, e^{At}\cdot x) = \left(1-\frac{1}{c\cdot r}\ \mathrm{dist}(x,B_r(0) ) \right)^+$

Fix $A \in \mathbb{R}^{m\times m}, c>0, r>0.$ Consider the function $f: \mathbb{R}_+ \times \mathbb{R}^m \to \mathbb{R}$ such that, for all $x \in B_r(0)$, it holds that $$f(t, e^{At}\cdot x) =...
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1answer
11 views

Growth rate of a function vs rate of change of a function

In lots of literature and article authors refer to the growth rate of a function, but nobody tells what it really is. Currently I should study math in english, but ...
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0answers
11 views

A problem involving intermediate value

This problem is motivated when I looked into Darboux’s Theorem. Suppose $I\in\mathbb{R}$ is an interval. Let $\mathfrak{P}(I) := \{f:I\to\mathbb{R}|(f(a),f(b)) \subset f((a,b))\}$. In other words, $\...
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0answers
65 views

counterexamples with complex function

I want to find counterexamples for the following "states": if $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad \forall x ,y\in\mathbb{C}.$$ prove or ...
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1answer
24 views

What would a rigorous proof that the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$ is not finite-dimensional look like?

So consider the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$. Intuitively, this vector space is not finite-dimensional. Indeed, $\mathbb{Z}$ and $\mathbb{N}$ are the same by "zig-...
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2answers
41 views

How can I prove $\text{Im } f = (0, 1]$ for $f(x) = 1/(1 + x^2)$?

I want to show that for the function $f : [0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = 1/(1 + x^2)$, we have $\text{Im} f = (0, 1]$ for $x \geq 0$. I think that I can do this by ...
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0answers
17 views

Finding factors s.t. f(x) > g(x)

I am looking forward for an answer to the following question: I have a function $f(x) = 1 - e^{-(x / \lambda)^k}$ . I want to find $\lambda'$ and $k'$ in $g(x)$ such that $g(x) < f(x) \in \mathbb{...
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1answer
44 views

Find a continuous function that is always less than or equal to its own integral

I am having difficulty coming up with examples of functions $f$ defines on $[0,1]$ such that $$f(t) \leq \int_{0}^{t} f(s) ds$$. I see that the is required to be that $f(0)=0$, and that the ...
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1answer
41 views

Surjective, not injective function [on hold]

Set $A$ is a set of positive odd integers.What is a function such that $f(A)$ is surjective (many to one) but not injective(one to one)?
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2answers
16 views

Do we have to include the value corresponding to the cancelled factor in the range?

Range of the function: $f(x) = \dfrac{x-2}{x^3 + 2x^2 - 4x - 8}$ is equal to ? a) $[0,\infty)$ b) $(0,\infty)$ c) $(0,\infty) - \dfrac 1 {16}$ d) $(0,\infty) - \dfrac 1 4$ ...
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3answers
244 views

Is $f: \Bbb{Z} \to \Bbb{Z}, f(x) = 3x + 6$ a bijection? Why? [on hold]

$f: \Bbb{Z} \to \Bbb{Z}, f(x) = 3x + 6$ Is $f$ a bijection? If no, explain why it isn’t. If yes, find an expression computing $f^{−1}(y)$ for $y \in \Bbb{Z}$. How to approach that question?
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1answer
22 views

Is there point in calculating a function zero if zero isnt in it's domain?

I was practicing function domains and function zeros For example, this function: $f(x) = x e^{\frac{1}{x}}$ It's domain is $\{x\in \mathbb{R} : x\neq0 \}$ It's function zero: $f(x) = 0$ $x e^{...
2
votes
1answer
15 views

Understanding functional composition example in Untyped $\lambda$-Calculus

The following equalities are shown as an application of $\alpha, \beta, \eta$-conversion found within Untyped $\lambda$-Calculus. I am unable to reconcile the first equality. Is the example missing a $...
0
votes
1answer
15 views

How to prove that function-value is zero before specific input?

I have been asked to prove that $$ f(x)=2x-ae^{-x}(x^{2}+1) \; \; \text{where} \;\; a>0 $$ will reach the value $$ f(x_{0})=0 \;\; \text{when} \;\;x_{0}<\frac{a}{2} $$ but have noe clear ...
1
vote
1answer
41 views

Evaluate $\lim_{x \to 0} \left( \frac{a \cos x}{a+ b\sin x} \right)^{1/x}$, where is the mistake?

I am not interested in the answer. I am interested in where I have made an error. I am to evaluate: $$\lim_{x \to 0} \left( \frac{a\cos x}{a+b\sin x} \right)^{1/x}$$ First, simplify $$\lim_{x \to 0}...
3
votes
1answer
33 views

$D \subseteq \mathbb{C}$ so that $f(z):=z^2$ is diffeomorphism

$f:\mathbb{C} \to \mathbb{C}$ $$f(z):=z^2$$ How can I find the largest possible open set $D \subseteq \mathbb{C}$ so that $f$ is a diffeomorphism and determine $f(D)$? I know that $f(-z)=f(z)$. My ...
1
vote
5answers
38 views

Find the domain and range of the function $f(x) = \frac{x}{x^2 - 16}$

$$f(x) = \frac{x}{x^2-16}$$ $$f(x) = \frac{x}{(x-4)(x+4)}$$ I can see that the domain is $\{x|x\neq \pm 4\}$ I'm not sure what to do for the range though. $$y = \frac{x}{(x-4)(x+4)}$$ $$x = y(x-4)...
3
votes
1answer
31 views

Classification of common functions types

I'm trying to classify functions types. The problem is that english is not my native language and I'm not sure if I did it correct. According to my classification we could have the following functions ...
1
vote
2answers
27 views

$g(x) = 1- \sqrt{x+2}$ For which $x \in \Bbb R $ is $g(x)$ positive?

What happens when $x < -2$ ? Does the whole square root term just "disappear" which leaves us with 1 which is positive and thus the answer to the question is $x\le-1$? Or do we have to constrain ...