Operating machines

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Age

16 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

My friend has two function machines. The first, called RECIPROCAL, takes as an input any function $f(x)$ and returns another function $1/f(x)$. The second, called PRODUCT, takes two functions $f(x)$ and $g(x)$ and returns a new function which is their product $f(x)g(x)$.

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.

Teachers' Resources

Why do this problem?

This problem gives an insight into the concept of operators and will give a great deal of practice in integration and differentiation, reinforcing the concept that they are inverse operations. It also reinforces ideas of closure, so important in more advanced topics in algebra.

Possible approach

This is a very hands-on problem, although it will initially require careful reading to understand the meaning of the mathematical operations involved.

Once the meaning of the problem has been understood, students should be encouraged to explore the possibilities generated by repeatedly applying the operations, look for patterns and then try to explain the meaning of the patterns.

Students should be encouraged to look for complete sequences of functions found by repeated application of the operators, such as the set of functions $\frac{x^n}{n!}$ where $n$ is a natural number.

They can choose any familiar function to begin with. Some, such as $\sin(x)$, will lead to interesting finite sequences, whereas others, such as $\ln(x)$, will lead to problems. Others, such as $\sqrt{x}$ will clearly lead to an infinite sequence of possibilities, although students might find the task of writing an algebraic expression for the general term a challenge.

To start students thinking about the more challenging combinations of operations, first suggest looking at combining DIFF and RECIPROCAL, starting from the function $f(x) = x$. Many interesting properties emerge: for example, students might be able to spot how to generate $2^n x$ for any value of $n$.

Key questions

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 $f(x)=x$?

Possible extension

There is plenty of extension built into the problem. The better students should really focus on finding complete sets of possibilities, and where possible find algebraic expressions for the terms in such sets. They might also look at the effects of starting with a pair of initial functions.

Possible support

Focus on the functions $f(x) = x^n$ for various whole values of $n$.

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