I am looking for some resources (books, Web sites, etc.) for teaching calculus students about inverse functions, using the ideas of codomain and onto functions (as well as one-to-one functions, of course). This is the way that most mathematicians think about inverse functions, and I don't think it is that difficult. I have looked at some popular calculus texts and some Web sites and they seem to completely ignore the concepts of codomain (or "target") of a function and onto functions. Maybe students can get by with this in Calc I, but in more advanced courses they will either have trouble or their instructors will have to waste time teaching them these basic concepts.

More info: I'm teaching Calc I. The books I seen don't mention codomain at all, they just define domain, range, and one-to-one function, then they define the inverse of a one-to-one function $f$ as the function $f^{-1}$ whose domain is the range of $f$, etc. etc. (you all know the rest)

EDIT: Since all the functions we'll be inverting this semester have codomain $\mathbb{R}$, I'm considering just doing it the way the calc books do it. My students may not see any purpose to defining a codomain, and I'm afraid that expressly giving a codomain for each example might confuse them because they have not seen real-valued functions treated that way before. Still, I think that when calc books define the idea of a function early in the book, they should introduce the important and simple concept of codomain.

  • $\begingroup$ Are you teaching calculus 2? $\endgroup$
    – imranfat
    Sep 23, 2013 at 19:57
  • $\begingroup$ Calc I. I added to the question. $\endgroup$ Sep 23, 2013 at 20:26
  • 1
    $\begingroup$ How to Prove It: A Structured Approach, by D.J. Velleman $\endgroup$
    – Git Gud
    Sep 23, 2013 at 20:31
  • $\begingroup$ they ought to know inverses of it through precalculus. Just checked a book. In calc2 this stuff comes back when dealing with transcendental functions (First chapter of the calc2 course) I do find most books kind of short on exact terminology as far as codomain, bijection etc. But then, they don't have to know that to such a detail. If you can make them invert properly, know the interchange of x&y properties, you will be doing a heck of a job. $\endgroup$
    – imranfat
    Sep 23, 2013 at 21:56
  • $\begingroup$ @Git Gud: Thanks. I'll check it out. I may use it as a supplement in the future if I can't find a calculus book that does it the "right" way. $\endgroup$ Sep 23, 2013 at 22:52

1 Answer 1


It is not a resource per se, but I typically introduce students to inverse functions as follows (this can all be done in a 1hr lecture):

  1. I begin by quickly discussing some properties of binary operators (without using that terminology though) such as multiplication and addition. "Let's look at things which eat two objects and create a third. For example, addition takes in two numbers $a,b$, and produces a third $a+b$. Multiplication takes in $a,b$ and produces $a\times b$."

  2. I define the operative identity. Typically students are quick to catch on to what we are discussing with a few examples. "We say that $\mathrm{id}$ is the 'identity element' of an operator if acting against $\mathrm{id}$ does nothing. For example, when using addition we require that $a + \mathrm{id}_+ = a$ for all $a$. A more specific example is $2+\mathrm{id}_+=2$ or $-5 + \mathrm{id}_+ = -5$." Most students guess that $\mathrm{id}_+ =0$ as required. Similarly, some examples will quickly reveal to the student that $\mathrm{id}_\times = 1$.

  3. I comment at this point that while the additive inverse always exists, the multiplicative inverse does not always exist (since $0$ fails).

  4. I comment how function composition (from $\mathbb R \to \mathbb R$) is a binary operator, but emphasize that unlike addition and multiplication, it is not commutative. Some quick examples will convince the student that $\mathrm{id}_\circ(x) = x$.

  5. I define the inverse of a function with respect to function composition, emphasizing that the inverse must be a function. I also emphasize at this point the difference between $\frac1{f(x)}$ and $f^{-1}(x)$.

  6. I give plenty of examples. Primarily the functions $f(x) = x^2$ and $g(x) = \sqrt x$, since this often violates the students' intuition. It also allows me to demonstrate that the domain and codomain are important in the definition of a function, since by changing the domain from $\mathbb R$ to $\mathbb R_{\geq 0}$ we get proper inverses on $f$ and $g$. It also allows me to show that functions can have one-sided inverses.

  7. The remainder depends on the level of calculus, with the following being introduced only in more rigorous classes: I define injective functions, and (hand-wavingly) motivate how inversion amounts to reflection about the line $y=x$, so injectivity ensures uniqueness of the inverse image.

  8. I define a surjective function, which again is (hand-wavingly) used to motivate well-definedness of the inverse function on its domain.

  9. I define bijective functions, and intuit that these are the functions which will have inverses.

  10. I show that inverses interchange the domain and codomain, making rigorous the previous two points.

  11. I show that a function is injective iff it has a left-inverse, and surjective iff it has a right inverse. Thus bijections have two sided inverses. It is then easy to prove that those inverses are identical and unique.

  12. * I show the stereographic projection of $S^1 \to \mathbb R$. It is simple and only uses the most basic properties of geometry and algebra, but shows that there is a bijection between the circle without a point, and the real line. I can then argue how poking a hole in the circle gives me $\mathbb R$, or how by adding a point '$\{\infty\}$' to $\mathbb R$ I can create a circle. The keen students usually find this pretty cool.

  • $\begingroup$ Though it doesn't really answer my question, I accepted your answer, because I appreciate all the work you put into it. Also MSE was complaining that I hadn't accepted an answer. If you can do all this in an hour and your students understand it (especially the abstract parts 1-3), then you and/or your students are more talented than me and my students. $\endgroup$ Oct 11, 2013 at 1:37

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