1st Six Weeks Project For your project this six weeks you will be completing the following problem set regarding combinations, compositions, and inverses of functions. The goal of this project is to connect the various ideas about functions we have been studying to both the equation and the graph of the function. You have been given the equation and the graph for two functions. Whenever possible, you should connect answers to the questions below to both the graph and the equation. All answers must be justified using both the graph and the equation for full credit. Please be sure to explain where all values came from. You may use your calculator or a graphing program like sketch pad to find some numerical answers, but you need to explain that that’s what you did. (if you would like to use sketchpad, you need to set up a time with me after school to come use one of the laptops.) You may discuss these questions with your classmates, but be careful that your final product represents only your own work. For each question below, give the correct answer and a justification. The correct answer is worth one point, the justification may be worth up to 3, depending on the quality of the justification. Please write or type your answers neatly using complete sentences on a separate sheet of paper. Provide the number and the question, then your answer and justification. Your answers should be put in a file folder and secured with a paper clip. Your name and class period should be written neatly on the outside of the file folder. For the graphs of the functions f and g which are attached 1. What is the domain and range of f? 2. What is the domain and range of g? 3. What is the domain of any combination except division? Why? 4. What additional domain restrictions would there be for for
g (x) ? f f (x) ? Why? What about g
5. What would be the range of the sum of these two functions? Justify your answer. 6. Find the composition f(g(x))
7. What is the domain of the composition f(g(x))? Remember that the range of the inner function has to fit the domain of the outer function. Is it immediately apparent from the expression you got above? 8. Try to take the composition for a point not in the domain using the graphical methods we discussed. What do you notice? How does the expression “the range of the inner function has to fit the domain of the outer function” manifest with graphically taking the composition for f(g(2))? 9. Find the composition g(f(x)), what is it’s domain? Is the domain apparent from the expression you got for the composition? What happens if you try to take g(f(2)) graphically? What does this correspond to algebraically? 10. Plot a few points on the composition f(g(x)). How can you verify that these points are accurate, using the equations? 11. Repeat each of the following for the graphs of r and s. How is the process the same? How is the process different? 12. Write two equations that when you take the composition the domain is the same as the domain of the inner function 13. Write two equations that when you take the composition the domain is different than the domain of the inner function.