# Explaining Closure to “Basic” Algebra 2 Students

I am currently introducing function operations to my basic level Algebra students and it has been semi-disastrous. The biggest problem I have with them is the notation that $$(f+g)(x)=f(x)+g(x)$$ I have been trying to explain that the reason for the notation on the left is the fact that when you add two (polynomial) functions, you get a polynomial function in return. Hence the $(x)$ on the right is indicative of this idea of function closure under the operations of addition, subtraction, multiplication, division, and composition. The district I teach at wants kids to understand the notation. If I just make them do $f(x)+g(x)$ they understand, but, with products for example, if I do $(fg)(x)$ they are lost. Any ideas how to appoach the subject with the basic learners?

• I used to do this by "$f$ is a machine and $g$ is a machine, and we can put these two machines together in several ways $\ldots$" Then discuss adding the machines, multiplying the machines, applying one machine to another machine's result, etc. My memory of textbooks at the U.S. college algebra level is that this abstract stuff came at the beginning of working with functions, and I often found it easier to do this after students had plenty of concrete practice with functions and function notation, so I often waited until the logarithms and inverse functions sections to talk about it. – Dave L Renfro Feb 9 '15 at 19:48
• I find it in very bad taste to go down to this level of elementary-hood and use the same symbol, $+$, with two different meanings. Who knows? Maybe this clarification will help out. – Git Gud Feb 20 '15 at 10:42

For your students, they are defined that way.

Particularly in the lower level algebra, \begin{align*} (f+g)(x) &= f(x) + g(x)\\ (fg)(x) &= f(x)g(x) \end{align*} are taken to be the definition of usual multiplication and addition. Simply stated, this notation can be used to talk about the names $f+g$ and $fg$ of the new functions created by adding or multiplying them. And whenever they see the above, that's what they need to write. It is similar for them with composition being $(f \circ g)(x) = f(g(x))$ Getting into the functional ring theory will only serve to confuse even though it is interesting to us.

• I just wanted to add at as a comment, these maps can be thought of as an evaluation map from the ring of continuous functions on a compact, hausdorf (for ease) space $C(X)$ to a point $x \in X$. The definition above is the result of the map being a ring homomorphism. – Chris C Feb 9 '15 at 19:55
• I used to (I no longer teach) emphasize that $(f+g)(x)=f(x)+g(x)$ was NOT an application of the distributive law, but I don't know to what extent this may have cleared up a possible confusion rather than raise an issue that never would have occurred to students in the first place. And for what it's worth, I've had a course out of Gillman/Jerison, and two of the three people on my Ph.D. committee worked in this area (and each have had several Ph.D. students in this area, but my work was with someone else in another area). – Dave L Renfro Feb 10 '15 at 14:59

Are the kids familiar with vectors and coordinates? If so, consider two vectors $\vec a,\vec b\in\mathbb R^3$, say. If $\vec c=\vec a+\vec b$, they probably know that $\vec c_i=\vec a_i+\vec b_i$. (It is convenient here to denote the $i$th component of $\vec c$ by $\vec c_i$ instead of $c_i$.) But this means that $\vec c_i=(\vec a+\vec b)_i$, right? Now it might make sense to write $\vec a_i+\vec b_i=(\vec a+\vec b)_i$.

If this sounds reasonable to the students, try to explain what is going on: the vectors $\vec a$ etc. are the objects that we want to study, and they are determined by their coordinates.

The same happens with polynomials. The things we wish to study are polynomials, say $f$ and $g$ (like we had vectors $\vec a$ and $\vec b$ before). If you think of polynomials like vectors (they in fact are vectors, but you might not want to mention this depending on what you want to do), then $f$ is the vector and $f(x)$ is the component. Make sure your students understand the difference between $f$ and $f(x)$ – and that of $\vec a$ and $\vec a_i$.

Now what might it mean to add two polynomials $f$ and $g$? Well, it is defined by its "coordinates", and the analogy from the vector side is $\vec a_i+\vec b_i=(\vec a+\vec b)_i$. This translates to $f(x)+g(x)=(f+g)(x)$.

Alternative idea: Try to introduce vectors (if the kids are familiar with them) as functions. Write $a(i)$ or $\vec a(i)$ instead of $a_i$ or $\vec a_i$; after that bit of new notation the jump to polynomials (and other functions) might be easier.

Final remark: I think the key is making a difference between $f$ and $f(x)$. You have little hope before they mean different things.

• I appreciate the help, unfortunately we cover vectors in precalc here so my students haven't seen vector notations. (It is covered in Geometry briefly...but focused on in precalc.) – Eleven-Eleven Feb 10 '15 at 17:19
• @Eleven-Eleven, oh, thanks for the explanation. It goes the other way here in Finland; at least for me vectors and components came years before addition of functions so I kind of assumed it would be like that everywhere. The usefulness of my idea depends a lot on the curriculum. – Joonas Ilmavirta Feb 10 '15 at 19:03