Elementary questions about functions, notation, properties, and operations such as function composition.
33
votes
4answers
2k views
How do I define a bijection between $(0,1)$ and $(0,1]$?
How do I define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if ...
16
votes
1answer
538 views
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 ...
1
vote
1answer
373 views
continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
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 $a>0$ ...
16
votes
2answers
610 views
Is there a natural way to extend repeated exponentiation beyond integers?
This question has been in my mind since high school.
We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
3
votes
2answers
98 views
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 ys, since the function ...
4
votes
3answers
950 views
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 ...
4
votes
2answers
320 views
On sort-of-linear functions
Background
A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies
$$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$
$$ (2)\;\; f(\alpha x) = \alpha f(x) $$
for all $ x,y \in ...
2
votes
3answers
3k views
If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that…) [duplicate]
Possible Duplicate:
Injective and Surjective Functions
If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one given that $f$ is a function from A to B and $g$ a function from B ...
18
votes
9answers
2k views
How do you define functions for non-mathematicians?
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
33
votes
4answers
1k views
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 ...
8
votes
3answers
487 views
Proof of a simple property of real, constant functions.
I recently came across the following theorem:
$$
\forall x_1, x_2 \in \mathbb{R},\textrm{function, } f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto y; \ |f(x_1) - f(x_2)| \leq (x_1-x_2)^2 \implies ...
6
votes
1answer
206 views
Arithmetic function to return lowest in-parameter
Is there a mathematical function such that;
f(3, 5) = 3
f(10, 2) = 2
f(14, 15) = 14
f(9, 9) = 9
It would be even more cool if there's a function that takes ...
11
votes
6answers
918 views
In written mathematics, is $f(x)$a function or a number?
I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real ...
1
vote
3answers
163 views
Is $f^{-1}(f(A))=A$ always true?
If we have a function $f:X\rightarrow Y$ where $A\subset X$, is it true to say that $f^{-1}(f(A))=A$?
3
votes
6answers
416 views
Writing a function $f$ when $x$ and $f(x)$ are known
I'm trying to write a function. For each possible input, I know what I want for output. The domain of possible inputs is small:
$$\begin{vmatrix}
x &f(x)\\
0 & 2\\
1 ...
