The Incomplete Treatment of Functions in HSC Mathematics

What Does The Syllabus Say?

The first port of call for all students and teachers when learning or teaching HSC Mathematics is of course the Syllabus. Here is the excerpt from the Syllabus (Page 31) about how a function is defined:

define and use a function and a relation as mappings between sets, and as a rule or a formula that defines one variable quantity in terms of another
- define a relation as any set of ordered pairs \( (x,y) \) of real numbers
- understand the formal definition of a function as a set of ordered pairs \( (x,y) \) of real numbers such that no two ordered pairs have the same first component (or \(x\)-component)
use function notation, domain and range, independent and dependent variables

The suggestion from the Syllabus to formally define a function as a set of ordered pairs of real numbers is neither formal nor is it the correct definition of a function:

Restricting the definition of a function to ordered pairs of only real numbers as opposed to elements from any set of objects – is incorrect.
The correct part of the definition is the notion of ordered pairs that links the independent variable \( x \) with the dependent variable \( y \), and the first component only occurs once for all ordered pairs. This is fine.
The so-called formal definition lacks any terminology such as domain, codomain and range. In fact, the Syllabus dot-points lack any mention of the term “co-domain” which is a worry if the course aims to prepare students for university, and these fundamental concepts in the definition of a function is used from Day 1 of any MATH1001 course.

In the next dot-point, the Syllabus mentions the use of function notation, domain and range, independent and dependent variables – again, this is missing any mention of the word “co-domain”.

Function Notation in the HSC Syllabus is usually taught like this:

\[ f(x) = \sqrt{x-4} \]

Here, the Domain and Range of the function is implicitly defined by the maximal subset of the real numbers for which this function is well-defined (i.e. behaves correctly). Unfortunately, this is not the usual practice once out of high school and implicit definitions are not always a good idea in a discipline of knowledge as rigorous and precise as Mathematics.

What Does The Glossary Say?

When one looks up the Syllabus Glossary on the definition of a function, it’s as though two completely different teams of people wrote the Syllabus and the Glossary separately. Here is how the Glossary defines a function:

A function \(f\) is a rule that associates each element \(x\) in a set \(S\) with a unique element \( f(x) \) from a set \(T\).
The set \(S\) is called the domain of \(f\) and the set \(T\) is called the co-domain of \(f\). The subset of \(T\) consisting of those elements of \(T\) which occur as values of the function is called the range of \(f\). The functions most commonly encountered in elementary mathematics are real functions of a real variable, for which both the domain and co-domain are subsets of the real numbers.
If we write \(y=f(x)\), then we say that \(x\) is the independent variable and \(y\) is the dependent variable.
Mathematics Advanced Stage 6 Syllabus 2017 – Page 70

What the?

There’s technical terms in here such as domain, range, co-domain. There’s also the use of universal quantification of “each element \(x\)”. There’s mention of elements in sets, and uniqueness. The mathematical language used here makes the definition clear, with no room for ambiguity or contradiction.

The Syllabus’ formal definition pales in comparison to the one found in the glossary! Both are in the same document published by NESA!

The glossary definition also mentions that most functions encountered in elementary mathematics are real functions of a real variable – as opposed to all of them as incorrectly implied in the Syllabus dot-point.

When I teach my students the definition of a function, this is the one I would use. However, they should note that the weaker and incorrect definition with no mention of co-domain in the Syllabus would imply that the understanding of the co-domain will not be assessed in the HSC.

I recall my first year of university back in 2010, having completed HSC Extension II in the year before: I was at a loss when the lecturer started mentioning the co-domain of a function as opposed to just range – judging from the murmurs and hands going up, I was not the only one confused in that lecture hall until I unlearned the HSC ‘definition’, and relearned the complete definition.

The Complete Notation

Let’s unpack the Glossary definition a little bit further. For any students reading this article, the notation introduced in this section will be useful in their preparation for any Mathematics units of study in university.

A function \(f\) is a rule that associates each element \(x \) in a set \( S \) with a unique element \( f(x) \) from a set \( T \).
The set \(S\) is called the domain of \(f\) and the set \(T\) is called the co-domain of \(f\)…
Mathematics Advanced Stage 6 Syllabus (2017) – Page 70

To denote that there is a function \(f\) that has domain \(S\) and co-domain \(T\), we write this:

\[ f: S \rightarrow T \]

To denote that \(x\) is associated with a unique element \( f(x) \), we write this:

\[ x \mapsto f(x) \]

The \(\mapsto\) symbol is read as “maps to”.

So where does Range fit into all of this? The Glossary defines Range as this:

The subset of \(T\) consisting of those elements of \(T\) which occur as values of the function is called the range of \(f\).

In Set notation, this would be \( \{ f(x) | x \in S \} \). We can learn how these terms are used in the following example.

Examples

Consider the following function:

\[
\begin{align*}
f: \{1,2,3\} \rightarrow \mathbb{R}\\
x \mapsto x^2
\end{align*}
\]

The domain is the set \(\{1,2,3\}\), the co-domain is the set of real numbers \(\mathbb{R}\), and the range is the set \(\{1,4,9\}\).

Sometimes, the rule can’t be defined with an algebraic rule such as the following function.

\[
\begin{align*}
f: \{a,b,c,d\} \rightarrow \{1,2,3,4,5\}\\
a \mapsto 1\\
b \mapsto 3\\
c \mapsto 2\\
d \mapsto 5
\end{align*}
\]

In this example, the domain is the set \(\{a,b,c,d\}\), the co-domain is the set \(\{1,2,3,4,5\}\), and the range is the set \(\{1,2,3,5\}\).

Why Though?

Why is the more complete definition more useful? What advantage does it have over the Syllabus one?

As previously mentioned, the more complete definition prepares students for university much better. When they learn more technical definitions and applications in their first year that rely on a clear understanding of these concepts, such as Surjections, Injections, Bijections – and in later years: Homomorphisms, Homeomorphisms, Isomorphisms, Automorphisms, etc (no, they are not swear words) – students who have already learned the correct definitions will not be hung on the basic concepts that should have been covered at the high school level.

The application of functions is not limited to only real variables, but to wherever there is a relationship between any two types of objects. For example, the notion of something being countable is tied to bijective functions with a domain of the set of Integers \(\mathbb{Z}\). This goes on to defining what “countable” and “uncountable” infinities are – or you may have heard the terms “discrete” and “continuous” in your Statistics classes.

In computer programming, functions are used all the time where the input (domain) and the type of output (co-domain) is defined by the programmer. Rarely would one be able to work out the range, or need to care, of such functions.

If there is one pure mathematical reason to learn the correct definition it is this: rigour, proof and correctness is the foundation of logical thinking and all of knowledge. To model this correctly to students, a fundamental concept such as Functions should be taught correctly!