Elementary questions about functions, notation, properties, and operations such as function composition.
62
votes
6answers
2k views
Find a real function, $f$, s.t. $f(f(x)) = -x$.
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:
Find a function $f: ...
37
votes
2answers
576 views
Looking for a function such that…
There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is:
...
34
votes
7answers
2k views
Why is $\log(\sqrt{x^2+1}+x)$ odd?
$$f(x) = \log(\sqrt{x^2+1}+x)$$
I can't figure out, why this function is odd. I mean, of course, its graph shows, it's odd, but when I investigated $f(-x)$, I couldn't find way to ...
33
votes
4answers
2k 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 ...
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 ...
31
votes
11answers
2k views
How is $e^x$ read aloud?
My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x.
How do you read aloud $e ^ x$?
Is it:
e raised to x
e power x
e powered x
or e ...
30
votes
8answers
2k views
When to Stop Using L'Hôpital's Rule
I don't understand something about L'Hôpital's rule. In this case:
$$
\begin{align}
& {{}\phantom{=}}\lim_{x\to0}\frac{e^x-1-x^2}{x^4+x^3+x^2} \\[8pt]
& ...
30
votes
4answers
972 views
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The question is:
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?
Edit
I should ...
25
votes
3answers
413 views
Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?
The function
$f(x)=x+\sin(x)$
is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map.
...
22
votes
6answers
601 views
Is there a function with this property?
Is there a real function over the real numbers with this property $\ \sqrt{|x-y|} \leq |f(x)-f(y)|$ ?
My guess is no but can anyone tell me why?
This came up as a question of one of my collegues and ...
22
votes
2answers
440 views
How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?
How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible.
I proceeded by using the ...
20
votes
9answers
3k views
How do you define functions for non-mathematicians?
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
20
votes
4answers
376 views
$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?
I found the following relational expression by using computer:
For any natural number $n$,
$$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$
Note ...
20
votes
0answers
358 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
19
votes
4answers
363 views
“$f$ is a function from $A$ to $B$” vs. “$f $is a function from $A$ into $B$”?
When we say that
$f$ is a function from $A$ to $B$
is this different from saying
$f$ is a function from $A$ into $B$
I know what injective ("1-1"), surjective ("onto"), and bijective ...
