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 defined . 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 . Instead, consider the function defined . This function is both one-to-one and onto (bijective). Therefore it is invertible, with inverse defined .
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.