Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,005
questions
0
votes
0
answers
40
views
How to find period of an arbitrary periodic function?
Is there any known algorithm to find the period of a generic function which is known to be periodic?
The most direct approach would be to solve for $T$ the functional equation $$f(x+T)=f(x)$$ which is ...
- 265
0
votes
2
answers
63
views
Let $p(x)$ be a polynomial and $f(x)$ a line that is tangent to $p(x)$ at $r.$ Why must $p(x)-f(x)$ have a root of $r$ with even multiplicity?
I recently came across a problem which needed the fact that if $p(x)$ is a polynomial and $f(x)$ is a line with $f(x)$ being tangent to $p(x)$ at point $r$, then the function $p(x)-f(x)$ must have a ...
- 87
-5
votes
0
answers
62
views
Solve $[2x]=2[x]$ where $[x]$ denotes the greatest integer function [duplicate]
I need to solve $$ [2x]=2[x]$$ where $x\in\mathbb{R}$ and $[x]$ denotes the greatest integer function less than or equal to $x$
Obviously the trivial solutions are $x\in\mathbb{Z}$. Also some real ...
- 842
4
votes
1
answer
87
views
Does defining a function from $\Bbb R$ to $\Bbb R$ as a set of ordered pairs make sense?
In high school functions were treated variously as a machine with inputs and outputs or as a rule for creating an output for any input. But in university classes like abstract algebra and analysis, ...
- 313
0
votes
0
answers
16
views
Finding involute curve to tangentially join two circles?
I have a bit of a curly maths problem I am struggling to solve.
I am designing a splined connection which must be cut with a tool of 6mm diameter (we'll call that variable m), so there must be outer ...
- 1
-2
votes
0
answers
13
views
universal approximation of permutation equivariant function [closed]
Prove: Any permutation equivariant function can be decomposed into a permutation invariant function and a function that depends only on individual elements of the input
1
vote
0
answers
56
views
If 2 functions, say $f(x)$ and $g(x)$, are both $\in$ $\Theta(h(x))$, then does that mean $f \in \Theta(g)$?
I'm not sure how to exactly prove this formally with the definitions of Big-$\Theta$. Also, would this apply to Big-O or Big-$\Omega$, or no?
We have that (by combining the definitions of Big-O and ...
- 853
1
vote
1
answer
80
views
Are there terminologies for "one-to-one" but not "onto" functions, and "onto" but not "one-to-one" functions?
One-to-one (injective) functions are not necessarily not onto (not surjective).
Similarly, onto functions are not necessarily not one-to-one.
So, a function can be one-to-one and onto (bijective).
$f(...
- 5,458
1
vote
0
answers
43
views
How to combine the $4$-dimensions of spacetime into 1 dimension?
I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...
- 126
0
votes
0
answers
26
views
About a proof of the existence of a certain function. ("Topology Second Edition" by James R. Munkres)
In "Topology Second Editon" by James R. Munkres, the author proved the following proposition.
The explanation that a function is a box that takes an input and produces an output has ...
- 1,082
0
votes
2
answers
45
views
Find the domain of this function through analytical ways
Consider $f(x) = \ln(2 + xe^{x^2})$ in the domain $(a, 0]$ where of course $a < 0$.
I was wondering if it's possible to find $a$ via analytical methods, using theorems and definition of analysis in ...
0
votes
2
answers
60
views
Condition for a function to change its sign at $x=a$
Consider a situation in which we are only allowed to evaluate the value of a real function $f$ (and its derivatives) at a particular value $a$. Also, assume that the given function is $C^\infty$ ...
- 1,392
0
votes
1
answer
52
views
Difference between $y = |x|$ and $|y| = x$
The Modulus function or the Absolute Value function is generally taught at the pre-university level. Here's a question to test your basics.
Let $y = |x|$ and $|y| = x$ be two functions. The first is ...
- 139
43
votes
3
answers
5k
views
+100
How many steps are needed to turn one "a" into at least 100,000 "a"s using only the three functions of "select all", "copy" and "paste"?
Suppose that at the beginning there is a blank document, and a letter "a" is written in it. In the following steps, only the three functions of "select all", "copy" and &...
- 531
0
votes
0
answers
20
views
Number of functions with at least two consecutive values equal [duplicate]
Apologies for the poor title.
Let $N$ and $k$ be fixed. We can take $N\geq 6$ and $k\geq 5$ but I don't think this is important.
Where $[N]=\{1,2,\dots,N\}$ and similar for $[k]$, consider the set of ...
- 7,779