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Activity 8.1 – Composition of Functions Overview: This activity explores composition of function by using both functions defined by equations and functions defined in tables. It also has an application problem that illustrates composition of functions. Estimated Time Required: This activity should take about 30 minutes. Technology: None Prerequisite Concepts: Finding the composition of two functions numerically, graphically and symbolically Interpreting composition of two functions Discussion: Begin by reminding students that a function is simply a rule that takes an input and returns a single output. If you have two functions, then you can obtain a new function by the rule: 1. start with an input 2. obtain an output from the first function 3. use that output as the input for the second function to obtain a final output. Emphasize that the composition of g with f is (g f ) x g[f x ] and that it represents a new function h whose input values come from the domain of f and whose output values come from g. Questions involving composition using data tables tend to be quite difficult. Have students complete the noting that some cells cannot be filled in. Activity 8.1 – Composition of Functions 1. Let f x x 2 2x and g t 2t 5 . a.) If x 4 , find f 4 . Now find g f 4 . This is called composing g with f. b.) If we let h x g f x , find an explicit algebraic formula for h. c.) Now find a formula for d.) Is f g t . g f x f g t ? Why or why not? 2. Fill in the values in the table (some of the cells must be left blank). inputs 1 2 3 4 5 outputs for f 4 3 5 2 3 outputs for g 3 4 0 2 1 outputs for g composed with f outputs for f composed with g outputs for f composed with f 3. An oil tanker has sunk, and there is oil leaking out in a circular oil slick. Suppose that the radius of the spill t hours after the accident is given by the area of a circle is a function of its radius given by r f t 300t meters. Note that A g r r2 . a.) What does the function g composed with f represent? b.) Find a formula for the area of the oil slick t hours after the accident. c.) For how many hours has the oil been leaking if the slick has an estimated area of 10,000,000 square meters?