Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
2,755
questions
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votes
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Why $f(x) = \sqrt{x}$ is a function?
Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a ...
user93957
39
votes
2
answers
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Is there a Cantor-Schroder-Bernstein statement about surjective maps?
Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.
I was ...
- 2,231
34
votes
5
answers
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Domain, Co-Domain & Range of a Function
I'm a little confused between the difference between the range & co-domain of a function. Are they not the same thing (i.e. all possible outputs of the function)?
- 393
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5
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Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?
I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic?
for every two real numbers $\alpha,\beta,\alpha\lt\beta$,
$$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$
I ...
- 2,373
77
votes
7
answers
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Functions that are their own inverse.
What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
- 6,612
57
votes
4
answers
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Is composition of measurable functions measurable?
We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
- 593
33
votes
1
answer
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Characterising functions $f$ that can be written as $f = g \circ g$?
I'd like to characterise the functions that ‘have square roots’ in the function composition sense. That is, can a given function $f$ be written as $f = g \circ g$ (where $\circ$ is function ...
- 9,805
18
votes
5
answers
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Show that if $g \circ f$ is injective, then so is $f$.
The Problem:
Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions.
(a) Show that if $g \circ f$ is injective, then so is $f$.
(b) If $g \circ f$ is surjective, must $g$ be surjective?
...
- 3,193
6
votes
4
answers
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Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
Let $f: A\longrightarrow B$ be a function.
1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$.
2)Give an example of a function $f$, and sets $C$...
- 941
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votes
3
answers
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Prove $f(S \cup T) = f(S) \cup f(T)$
$f(S \cup T) = f(S) \cup f(T)$
$f(S)$ encompasses all $x$ that is in $S$
$f(T)$ encompasses all $x$ that is in $T$
Thus the domain being the same, both the LHS and RHS map to the same $y$, since the ...
- 199
3
votes
1
answer
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continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ [duplicate]
Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.
$g(0)=1$. If $a=g(1)$,then $a>0$ ...
- 69
41
votes
7
answers
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views
Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
- 797
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votes
4
answers
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What is $\arctan(x) + \arctan(y)$
I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$
which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by ...
- 3,517
71
votes
5
answers
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Nice expression for minimum of three variables?
As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive ...
- 16.4k
32
votes
4
answers
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Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
While reading about cardinality, I've seen a few examples of bijections from the open unit interval $(0,1)$ to $\mathbb{R}$, one example being the function defined by $f(x)=\tan\pi(2x-1)/2$. Another ...
- 22.4k
29
votes
5
answers
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Codomain of a function
At high school we were told that a function has a domain and a range, the function maps from the domain to the range. Such that the domain contains all and only the possible inputs and the range ...
- 1,525
22
votes
6
answers
3k
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Intuition behind Cantor-Bernstein-Schröder
The book I am working from (Introduction to Set Theory, Hrbacek & Jech) gives a proof of this result, which I can follow as a chain of implications, but which does not make natural, intuitive ...
- 4,875
15
votes
4
answers
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Dense set in the unit circle- reference needed
For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
- 931
12
votes
1
answer
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Construct a monotone function which has countably many discontinuities
I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
- 3,600
4
votes
1
answer
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how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$
solve
$ 3x^2+3xy-5y^2=55$
using number theory tools ,i have found the following
$\Delta=3^2+4(5)(3)=9+60=69$
$d=69,u=1$
$w_{69}=\frac{1+\sqrt{69}}{2}$
$O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...
- 1,032
51
votes
10
answers
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What is a function?
I have been quite confused by the definition of functions and their uses..
First of all can one define functions in a clear understandable way, with a clear explanation of their uses, how they work ...
- 783
45
votes
7
answers
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How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
How to get $f(x)$, if we know that $f(f(x))=x^2+x$?
Is there an elementary function $f(x)$ that satisfies the equation?
- 601
28
votes
18
answers
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Explaining Horizontal Shifting and Scaling
I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.
For example, ...
- 25.8k
21
votes
3
answers
5k
views
Infinitely differentiable function with given zero set?
For each closed set $A\subseteq\mathbb{R}$, is it possible to construct a real continuous function $f$ such that the zero set, $f^{-1}(0)$, of $f$ is precisely $A$, and $f$ is infinitely ...
- 219
21
votes
1
answer
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Can a function be increasing *at a point*?
From what I understand we say that a function is increasing on an interval $I$ if
$$
x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2).
$$
for all $x_1,x_2\in I$. I understand that some might call ...
- 43.8k
18
votes
7
answers
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How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
I'm doing exercise on discrete mathematics and I'm stuck with question:
If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \...
- 3,959
12
votes
4
answers
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Injective and Surjective Functions
Let $f$ and $g$ be functions such that $f\colon A\to B$ and $g\colon B\to C$. Prove or disprove the following
a) If $g\circ f$ is injective, then $g$ is injective
Here's my proof that this is true.
...
- 779
54
votes
6
answers
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Why is there no function with a nonempty domain and an empty range?
Let $A$ to be a nonempty set and $B= \emptyset$; then $ A \times B$ is a set. And let $F$ be a function $A \to B$. Then $F \subseteq A \times B$. By the axiom of specification, $F$ must exists (if I ...
- 1,027
48
votes
1
answer
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"Well defined" function - What does it mean?
What does it mean for a function to be well-defined?
I encountered this term in an exercise asking to check if a linear transformation is well-defined.
- 3,301
28
votes
6
answers
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Prove $\sin x$ is uniformly continuous on $\mathbb R$
How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon?
I proved geometrically that $\sin x<x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ...
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