Questions tagged [functions]
Elementary questions about functions, notation, properties, and operations such as function composition.
0
votes
0answers
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
votes
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.
$\...
0
votes
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 $0 < x < 1$.
...
0
votes
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 ...
2
votes
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} \...
0
votes
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 ...
0
votes
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://...
0
votes
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 ...
0
votes
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 -...
1
vote
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 $\...
0
votes
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 ...
5
votes
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} $$
I was thinking ...
0
votes
1answer
44 views
Divide and conquer: Why $F(0) = 0$? [on hold]
Reading algorithms. Fibonacci example.
I saw an example where it said $F(1) = 1$ and $F(0) = 0$. (Fibonacci)
Why $F(1) = 1$ and $F(0) = 0$? From where does it originate?
Thanks.
2
votes
3answers
61 views
Find the coefficient of $x^{10}$
We have been given the following function.
$f(x)$= $x$ +$x^2$ + $x^4$ + $x^8$ + $x^{16}$ + $x^{32}$ + ...upto infinite terms
The question is as follows:
What is the coefficient of $x^{10}$ in $f(...
-4
votes
0answers
35 views
Can someone explain how to go about solving this question? [on hold]
Let $m,n\in\mathbb N$. Define a mapping $f:\mathbb Z/n\mathbb Z\to\mathbb Z/m\mathbb Z$ by $f([a]_n)=[a]_m$.
(a) Prove that if $m|n$, then $f$ is a well-defined function. (That is, prove that if $[...
-4
votes
1answer
33 views
help with functions? [on hold]
Let $f: A \rightarrow B$ be a function and let $A_1,A_2 ⊆ A$. Prove that $$f(A_1 \cup A_2) = f(A_1) \cup f(A_2)$$
0
votes
1answer
37 views
Is $\frac{\partial^2_xf \partial_yf}{\partial_xf}$ a total derivative?
Say $f$ is a function of two variables $x,y$. Is $\frac{\partial^2_xf \partial_yf}{\partial_xf}$ a total derivative? Or can we show $\int dx dy\frac{\partial^2_xf \partial_yf}{\partial_xf}$ vanishes?
-1
votes
1answer
53 views
how to prove a function is injective iff it is surjective [on hold]
Let X and Y be nonempty sets and suppose f: X→Y is a function. Define a new function F: P(Y)→ P(X) by F(B)=$f^{-1}$(B). Prove that F is injective if and only if f is surjective.
Note: P(Y) and P(...
0
votes
1answer
12 views
How do I find the equation for any given function?
If I am given a graph with labeled points, and I’m asked to find the equation that represents the given function, how can I do so? Just by looking at points it would be nearly impossible, and trying ...
3
votes
1answer
37 views
Showing a group does not have a faithful finite-dimensional representation
I am having some difficulty with this problem. We are asked to show that there does not exist any monomorphism to GL(n, $\mathbb{C}$). The group in the domain is defined as follows:
Let G = $(P(\...
2
votes
3answers
47 views
Evaluate $\lim_{x \to 4} \frac{x^4-4^x}{x-4}$, where is my mistake?
Once again, I am not interested in the answer. But rather, where is/are my mistake(s)? Perhaps the solution route is hopeless:
Question is: evaluate $\lim_{x \to 4} \frac{x^4 -4^x}{x-4}$.
My ...
1
vote
2answers
46 views
The inverse function of $f(x)=\sqrt{x^2+1}+2x$
Find the inverse function of $f(x)=\sqrt{x^2+1}+2x$ with $x \in \mathbb{R}^+$.
2
votes
2answers
19 views
Directional derivative and composite functions
We have a function $f(p_0)$ with $p_0 \in \mathbb{R}^n$, and the vector $\vec{v}$ with $||\vec{v}|| = 1$. The derivative of $f(p)$ to $\lambda$ in the point $p = p_0 + \lambda \vec{v}$, calculated for ...
1
vote
2answers
74 views
Just find number of its roots: $\sin{x}-x\cos{x}=0$
There is an other equation here:
$$\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}$$
It is in a way that we know ...
1
vote
0answers
36 views
Give example of The function is continuous and its absolute value is constant and is f not constant? [on hold]
Give example of The function is continuous and its absolute value is constant and is f not constant ? In complex C
I think it is sin x
It is true or not?
0
votes
1answer
35 views
the difference of functions tends to zero
If for any $\epsilon >0$,there exists $a_0$(depends on $\epsilon$) such that $f(a_0)>1-\epsilon$,where $f$ is a real function.We also know that $f(a_0)\leq 1$,can we deduce that $f(a_0)^4=f(a_0)...
0
votes
1answer
53 views
Simplify $x^n + x^{n-1} + … + x^1 + 1$ [duplicate]
Can I somehow simplify $x^{n-1}+x^{n-2} + ... + x^{2} + x^{1} + 1$?
I would like to have an explicit formula for that sum, but could not figure out a way to do so.
Could you help me? Thanks!
0
votes
0answers
29 views
Generalizable combination of combinations ratio equation
I have been trying to solve the following problem for a few days and any help or direction towards combinatorics literature would be appreciated.
Given a starting sequence of length n, an index ...
1
vote
2answers
32 views
How to show the image of $h : (0, 1) \rightarrow \mathbb{R}$ defined by $h(x) = 1/(x^2 + 8x)$ is equal to $(1/9, \infty)?$
How to show the image of $h : (0, 1) \rightarrow \mathbb{R}$ defined
by $h(x) = 1/(x^2 + 8x)$ is equal to $(1/9, \infty)?$
Hi, I want to show that the image of $h : (0, 1) \rightarrow \mathbb{R}$ ...
0
votes
0answers
10 views
How to isolate y in tuppers self referential formula?
How would we go about isolating $y$ in the following inequality?
$${1\over2} = \mathrm{floor}(\mathrm{mod}(\mathrm{floor}(y/17)*2^{(-17*\mathrm{floor}(x)-\mathrm{mod}(\mathrm{floor}(y), 17))},2))$$
...
0
votes
2answers
27 views
Find $a,b,c$ if $f(4)=1, f(-2)=14, f(5)=-2$ and $f(x)=ax^2 + bx + c$
Would anyone please explain to me the steps needed in order to find $a,b,c$? I understand I must use elimination, but the answers I keep on getting are very unrealistic. Thanks in advance.
1
vote
0answers
34 views
Plot and “meaning” of the function $ f(t, e^{At}\cdot x) = \left(1-\frac{1}{c\cdot r}\ \mathrm{dist}(x,B_r(0) ) \right)^+$
Fix $A \in \mathbb{R}^{m\times m}, c>0, r>0.$ Consider the function $f: \mathbb{R}_+ \times \mathbb{R}^m \to \mathbb{R}$ such that, for all $x \in B_r(0)$, it holds that $$f(t, e^{At}\cdot x) =...
0
votes
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 ...
0
votes
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, $\...
4
votes
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 ...
0
votes
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-...
0
votes
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 ...
0
votes
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{...
1
vote
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 ...
0
votes
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)?
1
vote
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) $(0,\infty) - \dfrac 1 4$
...
-1
votes
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?
2
votes
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 ...
