# Inverse functions and the horizontal line test

I have a small problem with the following language in our Algebra 2 textbook. Do you see my problem?

Horizontal Line Test

If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.

Here’s the issue: The horizontal line test guarantees that a function is one-to-one. But it does not guarantee that the function is onto. Both are required for a function to be invertible (that is, the function must be bijective).

Example. Consider $f:\mathbb{R}\to\mathbb{R}$ defined $f(x)=e^x$. This function passes the horizontal line test. Therefore it must have an inverse, right?

Wrong. The mapping given is not invertible, since there are elements of the codomain that are not in the range of $f$. Instead, consider the function $f:\mathbb{R}\to (0,\infty)$ defined $f(x)=e^x$. This function is both one-to-one and onto (bijective). Therefore it is invertible, with inverse $f^{-1}:(0,\infty)\to\mathbb{R}$ defined $f(x)=\ln{x}$.

This might seem like splitting hairs, but I think it’s appropriate to have these conversations with high school students. It’s a matter of precise language, and correct mathematical thinking. I’ve harped on this before, and I’ll harp on it again.

## 2 thoughts on “Inverse functions and the horizontal line test”

1. Old folks are allowed to begin a reply with the word “historically.”

Historically there has been a lot of sloppiness about the difference between the terms “range” and “co-domain.” According to Wikipedia a function g: A -> B has B by definition as codomain, but the range of g is exactly those values that are g(x) for some x in A. Wikipedia agrees with you. That hasn’t always been the definition of a function.

When I was in high school, the word “co-domain” wasn’t used at all, and B was called the “range,” and {g(x): x in A} was called the “image.” “Co-domain” didn’t come into popular mathematical use until an obscure branch of mathematics called “category theory” was popularized, which talks about “co-” everythings. (Category theory looks for common elements in algebra, topology, analysis, and other branches of mathematics. It is an attempt to provide a new foundation for mathematics, an alternative to set theory or logic as foundational. That research program, by the way, succeeded.)

Now, what’s the inverse of (g, A, B)? Notice that I’m recognizing that a function is not a rule (g), but a rule, a domain, and a something. I agree with Mathworld that the function (g, A, B) has an inverse if and only if it is bijective, as you say. But note that Mathworld also acknowledges that it is fair to refer to functions that are not bijective as having an inverse, as long as it is understood that there is some “principal branch” of the function that is understood. (You learned that in studying Complex Variables.) OK, if you wish, a principal branch that is made explicit. We are allowed to say, “The sine function has an inverse arcsin,” even though to be more pedantic we should say that sin(x) on the domain (-pi/2, pi/2) has an inverse, namely Arcsin(x), where we use the capital letter to tell the world that we have limited the domain of sin(x). OK, to get really, really pedantic, there should be two functions, sin(x) with domain Reals and Sin(x) with domain (-pi/2, pi/2). Trick question: Does Sin(x) have an inverse? Pedantic answer: I can’t tell until you tell me what its co-domain is, because a function is a triple of things and you only told me the rule and the domain. Common answer: The co-domain is understood to be the image of Sin(x), namely {Sin(x): x in (-pi/2, pi/2)}, and so yes Sin(x) has an inverse.

What’s tricky in real-valued functions gets even more tricky in complex-valued functions. And to solve that, we allow the notion of a (complex) function to be extended to include “multi-valued” functions. See Mathworld for discussion. So when I say that sin(x) on the domain of Reals has an inverse, I might mean the multi-valued function arcsin(x) whose co-domain is sets of reals, not just reals.

Now here is where you are absolutely correct. Regardless of what anyone thinks about the above, engaging students in the discussion of such ideas is very helpful in their coming to understand the idea of a function.

There is a section in Victor Katz’s History of Mathematics which discusses the historical evolution of the “function” concept. You definition disagrees with Euler’s, and with just about everyone’s definition prior to Euler (Descartes, Fermat, Oresme). They were “sloppy” by our standards today.

“Sufficient unto the day is the rigor thereof.”

Let’s encourage the next Euler by affirming what we can of what she knows.