Ideas from Classroom Teachers for
See Navigating Through Algebra: Grades 9-12 (NCTM) for real-world types of activities for this topic.
Students need practice determining domain and range of step functions by looking at their graphs.
Finding the domain of a composite function can be difficult. The domain cannot always be determined by the simplified equation of the composite function.
Students should consider various representations of functions: algebraic (as equations), graphical, numerical (e.g., in tabular form), and verbal.
Relations that are not functions should also be considered.
Students should graph functions, explore and discuss the behavior they see, and come to conclusions. There are many such activities in Algebra in a Technological World (NCTM, 1995).
Students should investigate graphs of functions and their inverses to discover properties.
Function composition can be difficult for students. On the one hand, they may see the results better by using more complicated functions, but on the other, they seem to be less intimidated if we start with composing simpler ones (like a linear function with a constant function).
Inverses will be important as students learn how to solve different types of equations. I am assuming that at this point in the course students are familiar with the different types of functions. Depending on mathematics background, students at this level should be able to recognize graphs of standard functions: identity, squaring, cubing, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute value, and greatest integer functions.
A common theme throughout this year is how the study of algebraic functions is necessary to provide models for real-world situations. Investigations often begin with a set of data. If those data can be fit into an algebraic model, useful predictions can be made.
I take 3-4 days to introduce the basic function families. Students should be able to match a function with its graph and vice versa. We then define intercepts, maxima, and minima. (You may distinguish between absolute extrema and local extrema of a function, as well as extrema on a closed interval.) Domain and range are included in the discussions of the parent function and in the identification of extrema. I spend another 3-5 days on composition of functions; this leads into discussion of inverse function from a chart, from a graph, or from a formula. Ask students to determine from the original function whether its inverse will also be a function.
On inverse functions: Emphasis should be placed on illustrating inverses in numerical form (switch x and y in ordered pairs), algebraic form (switch x and y in the equation), and graphical form (reflection over y = x line). Also the definition of inverses relative to composition should be emphasized to illustrate why some functions are inverses (y = x^3 and y = x^(1/3)) and why some are not (y = x^2 and y = x^(1/2)), even though powers and roots are treated as inverse operations when solving equations.
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