Composition of Functions with GSP Name: Period Date: GENERAL GRAPHING INSTRUCTIONS Version 4 of GSP allows users to select multiple objects without using the shift key. Be attentive to what has been selected. When something does not work as it should, a first remedy is to deselect everything by clicking on blank space and then trying again. To construct a graph, go to Graph and click on either Define Coordinate System or Show Grid. A coordinate system should be displayed. GSP allows for functions to be graphed with parametric coefficients. For example, in order to graph y = mx + b, go to Graph| New Function. Under Values, click on New Parameter. In the Name box, type "m" and press OK. Then type ". x +" and go to Values, New Parameter, "b." By default, parameters are given the value of 1. One can change the value when creating a new parameter or by double-clicking on the parameter itself. COMPOSITION OF FUNCTIONS Type in the following two functions: f(x) = x2 - 3 and g(x) = 2x + 1. Once the functions are entered, make sure that only the functions are selected. Then click on Graph| Plot Function. The graphs should appear. Go to Graph|Plot New Function. Click on the first function, f(x), and f( ) should appear in the screen of the calculator. Click on the second function, g(x), and f(g()) should appear. Type "x" inside the innermost parentheses, and click OK. Does f(g(x)) = g(f(x))? Find h(x) such that h(g(x)) = g(h(x)) Find k(x) such that k(f(x)) = f(k(x)) Consider the following two functions: f ( x ) = 1 − x and g ( x) = 1 x We can compose them in two ways: f(g(x)) and g(f(x)) We can go further and compose these two new functions with themselves, and also with the old ones, in a number of ways. Keep composing these functions with new ones as they are generated and figure out simplified formulae for them in terms of the variable x. (Don’t forget to compose functions with themselves, like f(f(x)) .) You might think that more and more new functions will be generated. Surprisingly only a finite number of new ones get generated by composition, even though there may be many different ways of composing f and g to get the same function. Remember that two very different looking formulae may represent the same function. Tasks: A. How many distinct functions are there, including f and g themselves? B. List all of them. C. Show how each can be obtained from compositions of f and g. D. Show how you know you have found all of the possible compositions. E. Very carefully, state the domain of every composition you define. G:\School\2008-2009\Precalc\Investigations\Composition of Functions.doc