Functions and Invertible Functions: Response
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Egan J Chernoff, University of Saskatchewan (egan.chernoff@usask.ca)
Kseniya Garaschuk, University of the Fraser Valley (kseniya.garaschuk@ufv.ca)
Your teacher comes to class and says: “Today, we are going to discuss rings.” What is your first reaction? Puzzled, thinking about it, you want to ask: “What do you mean?” What could the ‘ring’ be? Maybe the teacher has rings of Saturn in mind, or the Ring of Fire (the earthquake-prone boundary of the Pacific); or maybe they meant a boxing ring, or a smuggling ring, or an engagement ring … maybe the ring from the Lord of the Rings. You are scratching your head … maybe it is about a commutative ring, a ring as the region between concentric circles, or a Borromean ring; or, a ring model as in a circular configuration of neurons in a network? Your teacher’s announcement is, to say the least, ambiguous!
This article is a response to the article “On invertible functions and on functions in general” published in the CMS Notes (June 2025) which highlights the importance of the initial definition of a function and the conditions needed for the invertibility of a function.
A bit of a background. As educators (in particular those of us involved in K-12 teacher education and tertiary education research) we routinely discuss situations which involve ambiguities, imprecise, unclear, or incomplete statements in mathematics. For instance, the exponent of −1 might denote a reciprocal, or an inverse function (e.g., the fact that \sin^{-1}x may mean \csc x or \arcsin x could be a source of confusion). The meaning of the term ‘multiplication’ depends on the context (product of real numbers, or functions, or matrices, scalar product, etc.), meaning that we have to be on alert and always ask ourselves ‘what is going on here?’. Bringing up such situations with our students could lead to productive discussions, which deepen their understanding. Arguing that 0.999\ldotsis equal to 1 involves asking ‘what does 0.999\ldots actually mean?’, and then using appropriate machinery (geometric series) to figure out the answer.
So far, so good – so what is the problem (i.e., why this response)? The question we are asking is this: “How far should we go beyond identifying ambiguities and discussing their pedagogical value, i.e., at what point do we move from questioning available definitions to choosing the one to work with?” To illustrate, we look at Tsamir and Tirosh (2025), who contrast the following competing definitions:
Definition 1: A function f is called increasing if f(x_1)\leq f(x_2) whenever x_1<x_2.
Definition 2: A function f is called increasing if f(x_1) < f(x_2) wheneverx_1<x_2
The authors use examples (and non-examples) to illustrate the differences between the two definitions, and this is where their narratives end. Of course, anyone teaching Calculus cannot afford to stop here – they need to select a definition in order to move on and continue developing mathematical concepts.
In mathematics education, there is a concept called horizon content knowledge, often abbreviated as HCK. The basic premise (adjusted to our argument) is that, when faced with having to make a decision (as above – Definition 1 or Definition 2?), we look forward, i.e., what happens later (in our course, or in a textbook) informs the decisions we have to make now. Since we would like to claim that increasing functions are invertible (which, looking at the farther horizon, is also a theorem in analysis), we must adopt Definition 2 (and this is what Calculus textbooks do).
When faced with ambiguities, we need to consider context and purpose. This means not only to ‘backward engineer’ the material or to decide what is important or relevant, but also to keep in mind who our audience is (mathematics and statistics majors, life sciences students, engineers, economics students, and so on).
Zazkis (2025) contrasts two definitions of a function, identified as the Ordered Pair definition (Halmos, 1960) and the Bourbaki Triple definition (Bourbaki, 1968), commenting that “while the similarity is evident, a no- table difference is explicit mention of domain and codomain in the ‘triple definition’ ”.
Halmos (1960) defines a function as a relation (p. 30); previously, talking about relations (p. 27), he defines two associated sets: the domain and the range of a relation. Thus, even though it is not explicitly mentioned, a function comes with its associated domain and range. The Bourbaki Triple definition conceptualizes a function as a triple (D,F,E), where D and E are sets and F is a subset of D\times E with the property that for every x\in D there is a unique y\in E so that (x,y)\in F.
The resolution of the issue raised (Which condition(s) guarantee the invertibility of a function?) can be reached by a careful reading of the two references. Halmos’s (1960) definition of a function as a relation means that a function is surjective (i.e., onto its range), and thus only one-to-one-ness is needed. Bourbaki’s definition says that E is the codomain (not the range), and thus both surjectivity and one-to-one-ness are needed. (We are sure that we glossed over some subtleties which might leave those in philosophy of mathematics unhappy.)
This discussion, so far (i.e., contrasting Halmos and Bourbaki definitions), is not something that we can conduct in most of our first-year university mathematics classrooms. However, as it does pose interesting questions, we now move this discussion to the context of Calculus.
A common definition of a function in Calculus is the following statement.
Definition 3. A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. (Stewart et al., 2021, p. 8)
The material presented prior to this definition (reading a definition in isolation is never a good idea) informs us that D and E are non-empty subsets of the set of real numbers. The set D is called the domain of f . Some textbooks do not explicitly name the set E the codomain of f (even though they use the notation f\colon D \rightarrow E); however, all Calculus textbooks define the range of f as the set f (D) of all values f (x) for x in the domain D of f .
From this definition it is clear that while f\colon D \rightarrow E does not have to be surjective, the function f\colon D \rightarrow f(D) is always surjective. Thus, the following statement holds.
Theorem 1. Assume that a function f\colon D \rightarrow E is one-to-one. Then f has an inverse function g defined on f (D). (modified from Stewart et al., 2021, p. 55)
When studying ambiguities about the conditions needed for a function to have an inverse function, Mirin et al. (2020, p.23) bring up the question on whether f(x)=e^x is invertible or not (as it is not surjective onto \bf R). With Theorem 1 in mind, one response is that the function f\colon {\bf R} \rightarrow {\bf R} defined by f(x)=e^x (being increasing and thus one-to-one) has the inverse function g (namely g(x)=f^{-1}(x)=\ln x) defined on f({\bf{R}})=(0,\infty).
Zazkis (2025) concludes the article by asking whether the functions g\colon {\bf R} \rightarrow {\bf R}, defined byg(x)=x^2, and h\colon {\bf R} \rightarrow [0,\infty), defined by h(x)=x^2, are equal. Of course, it is a matter of which definition we use.
Definition 4. Two functions f_1\colon D_1 \rightarrow E_1 and f_2\colon D_2 \rightarrow E_2 are said to be equal if their domains D_1 and D_2 are equal, their ranges f(D_1) and f(D_2) are equal, and f_1(x)=f_2(x) for all x in D_1=D_2.
Definition 5. Two functions f_1\colon D_1 \rightarrow E_1 and f_2\colon D_2 \rightarrow E_2 are said to be equal if their domains D_1 and D_2 are equal, their codomains E_1 and E_2 are equal, and f_1(x)=f_2(x) for all x in D_1=D_2.
Although Definition 4 is commonly used in Calculus, it is not always explicitly stated; for instance, it does not appear in Stewart et al. (2021). According to Definition 4, the two functions g and h are equal (we do not use the term ‘equivalent’ that Zazkis (2025) uses). However, this might not make us happy because h is onto, but g is not. If we wish to fix this, then we need to adopt Definition 5, which implies that g and h are two different functions.
There is another aspect of the definition of equal functions that is important – the equality of their domains. For instance, the functions \ln x^2 and 2 \ln x are equal only when viewed as functions with the domain D=\{ x \in {\bf R} \, | \, x >0\}. The statement \frac{x^2-1}{x-1}=x+1 is incorrect, unless we state that x \neq 1. The formula \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n holds only for |x|<1.And so on.
As we all know, mathematics never ends – we definitely do not view this article as a conclusion, but rather as an invitation to discuss these (and other) issues further.
References
Bourbaki, N. (1968). Theory of Sets. Don Mills, ON: Addison-Wesley Publishing.
Halmos, P.R. (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand Company.
Mirin, A., Milner, F., Wasserman, N., & Weber, K. (2020). On two definitions of ‘function’. For the learning of mathematics, 41(3), 21-24.
Stewart, J., Clegg, D., & Watson, S. (2021). Calculus, Early Transcendentals, 9th Edition. Boston, MA, USA: Cengage.
Tsamir, P. & Tirosh, D. (2025). Nonequivalent definitions: anecdotal incidents or an ordinary constancy? For the Learning of Mathematics, 45(1), 39-44.
Zazkis, R. (2025). On invertible functions and on functions in general Canadian Mathematical Society Notes, 37(3).
