1. LIMIT REVIEW AND COMPOSITE FUNCTIONS Lesson 11: How do we compose functions using graphs? Mini Theme: Famous Composers

2. Limit Review  Limit Existence:  The limit DOES NOT EXIST if any of the conditions below are not met: f(x) approaches a different number from the right side of x as it approaches from the left side of x. (piecewise function) f(x) increases or decreases without bound as x approaches c (rational function) f(x) oscillates between two fixed values as x approaches c

3. Who is this composer?  Widely considered one of the most influential musicians of the 20 th century.  The album Kind of Blue was his magnum-opus – it was commemorated by the House of Representatives in 2009 as a national treasure. It is also the top-selling album in the history of Jazz Music.

4. How do we know if a function is continuous?  Continuous function test  If the limit does NOT exist then the function is not continuous  A limit can exist, while the function is still not continuous  Test for Continuity of a Function:  If  Example:  Here the limit exists but the function is NOT continuous

5. Composite Function using Graphs  How do we compose functions with graphs?  Given:  Find:

6. Who is this composer?  Composter Francis Scott Key had a later family member who was named after him who became a famous writer.  Everyone in here has read this author’s book! F. Scott Fitzgerald

7. Composition Refresh:

8. CFU Answers

9. Who is this composer?  Despite becoming completely deaf at age 44, and dealing with ongoing illnesses, this composer continue to write beautiful music.  Beethoven

10. Average Rate of Change  What is the average rate of change for each of the graphs below?

11. Average Rate of Change = Slope  Find the average rate of change of f(x) on [-3,1]  f(x)g(x)

12. In case second day needed….  Example: Graph the below piecewise function:  Domain: ______Range: ________

13. Example continued:

14. CFU Follow up  Find the domain and range for f(x)  Does a limit exist at 2? Is the function continuous there?  Does a limit exist at 0? Is the function continuous there?