Operating machines
I start off with the function $f(x)=x$. Which functions can I make using the function machines RECIPROCAL and PRODUCT? Can you describe an entire theoretical set of such outputs? Prove your assertions. (Note that I have several copies of this function to hand). Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different
initial functions?
I have two operator machines. The first, called DIFF, takes any input function and returns another function which is its derivative. (If the derivative function does not exist, then DIFF returns the original function). The second, called INT, takes any input function and returns the implicit integral (i.e. no constant of integration) of the original function (if this integral does not exist then
INT returns the original function).
If I again start off with the function $f(x) = x$, what set of outputs can I make using DIFF and INT? Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different initial functions?
Can you find any initial functions which yield a finite set of possibilities under RECIPROCAL and PRODUCT or under DIFF and INT?
What happens if you begin with $f(x) = x$ and use any combination of RECIPROCAL and DIFF?
Explore the possibilities when you can use different combinations of RECIPROCAL, PRODUCT, DIFF and INT for various choices of initial function. Can you find any interesting sets of outcomes?
See also the problem Calculus Countdown
NOTES AND BACKGROUND
In mathematics and science, the concept of differential and integral operators is very important, with applications ranging from quantum chemistry to analysis of waves. The simplest examples of these are given in this problem under the guise of DIFF and INT.
Have you carefully read the question?
What happens if you repeatedly apply DIFF to $f(x)=x$?
What happens if you repeatedly apply INT to $f(x)=x$?
What happens if you apply DIFF several times followed by INT several times?
What functions might you try instead of $f(x)=x$?
Part 1: Using RECIPROCAL and PRODUCT
Starting with $f(x)=x$, $x^n$ is possible for all integers n, and there are no other possibilities. The same set of possibilities can be created by starting with $f(x)= \frac{1}{x}$. No other starting function will yield the same set.
Part 2: Using DIFF and INT
Starting with $f(x) = x$, it is possible to create $0$ and $\displaystyle{\frac{x^n}{n!}}$ for any non-negative integer $n$
Starting with any function from this set, it is possible to generate all others using DIFF and INT.
Part 3:
Functions yielding a finite set of possibilites under DIFF and INT are $e^x$, $\sin x$ and $\cos x$
$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} e^x = \int e^x = e^x}$
$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} \left( \sin x \right) = \cos x
\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left(\cos x \right) = -\sin x
\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\sin x \right) = \cos x
\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\cos x \right) = \sin x}$
The sequence then repeats.
There are others, for example $e^{-x}$ yields two possibilities, and $\sinh x$ and $\cosh x$ also yield two possibilities.