# How Do You...Composite Functions

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

### Objective

Students should become comfortable with function notation. Students should see how functions referenced by their notation can be combined through addition, subtraction, multiplication, and composition. Students relate these actions to perimeter, area, and volume.

### Progression

When they reach Pre-Cal, students should have seen f(x) notation for referring to more complicated expressions. I review this notation and what it means to find f(2). Usually all students are ok with this.

First, addition and subtraction. I generate two expressions, say f(x) = 3x + 7 and g(x) = 10x - 2. I pose the question: what do you think f(x) + g(x) would mean? Most are able to arrive at the answer that they need to combine like terms to arrive at f(x) + g(x) = 13x + 5. Subtraction follows a similar progression. They'll practice with a few examples I put on the board.

Next day we will discuss multiplication. I return to f(x) = 3x + 7 and g(x) = 10x - 2 and pose the question: what do you think f(x) * g(x) would mean? There is more struggle here. Some will say 30x - 14 is the answer. A few who thought about it longer will step in and say the answer is more complicated than that. At this point we debate methods. I offer the area model. I offer distribution (arrows flying everywhere). We arrive at 30x^2 + 64x - 14 either way. Students will have a preference. I make no endorsement of either.

I throw a little curveball. Define f(x) = (x+2)^2 - 10 and g(x) = 10x - 2. Any operations would require us to expand the binomial in f(x) first. Often students forget the "-10" there on the back when simplifying f(x), a good talking point.

Students will spend a day practicing addition, subtraction, and multiplication with a set of defined functions. A few of them will require them to expand a binomial first.

Here's an idea:

I don't explicitly mention scaling. Most can easily figure out what I'm after with 2*p(x).

We then work on the mechanics of f(g(x)) and then work another problem set of multiplication and composition. The activity below is designed to help put some context to f(g(x)).

### Activities

Ok, that's the mechanics part. Now for something more tangible. Students will find the logic here a bit of a stretch, the trouble of knowing how to multiply f(x) and g(x) without any thought as to where they came from or why we would bother.

Show the Dan Meyer tank filling video:

I simplify the tank as a cylinder, but you don't have to. Area of an octagon is not as readily retrievable. We work towards a definition of V(t), the volume of the water in the tank. We discuss the volume of a cylinder and define A(r), the area of the top, and h(t), the height of the water. V(t) is the result of A(r) * h(t).

The idea stretches further if you talk about V(h(t)) since the area of the top is a constant. They get weirded out by this idea because it looks overly complex (and kind of is). I have screwed up this part.

As a follow up, I will relate addition to finding the perimeter of a figure and relate multiplication to finding the area.

The idea is not to convince them that this stuff is magically more real, just to show them how more sophisticated notation can be used for a problem they recognize.

### Assessment

Students are assessed on the mechanics and context of these problems. For context, students are given some functions defined as the height and width of an unknown figure and they have to illustrate the scenario. It is tested across three SBG topics which takes three tests to complete.

Posted