The appropriateness of when to introduce the function concept varies depending on the views of the individual instructor and the student population one is working with. I value the importance of the function concept and notation, but feel that teaching the function notation and the “vertical line test” without follow-up material relating to curve analysis (max, min etc.), composition and inverse, and transformation of functions is of questionable value particularly in the case of a gen. ed. math course that is to be taken by a broad range of students.
I've found that it is possible to discuss many topics related to the behavior of polynomial, rational, exponential and logarithmic functions without the additional two or three days of material required to introduce the function notation to a gen. ed. student population.
Given time, I like to cover functions by introducing the basic notation, followed by curve analysis in which the concepts of increasing and decreasing behavior, as well as maximum and minimum values are discussed. I then like to address function composition and inverse, the transformation topics (which can be somewhat tricky for students) and finish with a unit on the application of functions to a range of optimization problems common in many Calculus courses. I typically have the students use the TI 83/84 calculators to find the maximum and minimum values that will be found algebraically in a Calculus course.
So, I typically spend four to five weeks covering
2) increasing, decreasing, max, min
3) composition, inverse
In the application and modeling topics, I try to group the problems so that the students can see them as three or four separate types of problems rather than having them jumbled together. I find that students have enough trouble with mathematical modeling that throwing all the different types of optimization problems together can be fairly confusing.
The major types of optimization we cover are analytic geometry, geometrical shapes/cost, pythagorean/cost and pythagorean/rate-time-distance.
A typical analytic geometry problem would be something like the following:
Find the point (x,y) in the first quadrant on the curve f(x)=x^2-5 that is closest to the point (4,1).
Express the distance of the point on the curve (x,y) from the point (4,1) as a function of x. Find the x and y values of the point on the curve that is closest to the point (4,1) and determine how far it is from this point.
This problem involves using the distance formula d=SQRT((x1-x2)^2+(y1-y2)^2) in relation to the given points from the problem. In other words the distance function would be d=SQRT((x-4)^2+(y-1)^2) – but, the question asks for this function to be in terms of x, so we should then write it as d(x)=((x-4)^2+(x^2-5-1)^2) since in the original function y=x^2-5.
Once this is written in terms of x, it can be graphed on the TI 83/84 and the minimum value of that curve can be found. What I often find is tricky for the students is keeping straight in their heads the difference between the original function f(x)=x^2-5 and the distance function d(x)=((x-4)^2+(x^2-5-1)^2).
This is why I ask for the x and y values of the point on the curve and also how far away from the curve this point is. The answers here are x=2.51 y=1.3 (appx) and the minimum distance d=1.52 (appx).
Since the Sullivan text I've been using has good material on this topic, I don't have the extensive supplementary problem sets that I do for the previous topics. Instead, here's a link to a sample Pre-Calculus mid-term. I'm thinking about splitting this into two tests this Winter - that is, having the applications be a separate test.
The newer textbooks generally have good material on functions, although they all typically separate the sections on composition and inverse to include these with the chapter on exponents and logarithms. In the Sullivan text, Chapter 3 contains some solid content on functions, including a good selection of application problems in section 3.6. The section on composition and inverse functions is in sections 6.1 and 6.2 – I usually cover this at the same time as the Chapter 3 sections.
The Coburn text covers functions in sections 2.4-2.8, with a section on inverse functions in section 4.1 (with the exponents and logarithms). The Coburn book is particularly weak on the standard application problems, although they do include applications on variation in section 3.8 and linear and quadratic regression in a special section following Chapter 2.
The Brown/Robbins text covers functions in Chapter 4, with a good variety of applications in section 4-4. The Dolciani text covers only the basic ideas of functions and mixes these topics in with material on rational roots in Chapter 6.