1. APC Unit 4 Composite functions Inverse Functions

2. Warm-up  Reflect on Your performance so far this year  And on this latest Test  What should you do differently?  What should we do differently?  How do we make class time more productive

3. 5.1 Composite Functions  (f o g)(x) = f(g(x))  “f composed with g of x”  “f of g of x”  Evaluate or substitute from the inside out  That is, evaluate the inner function, g(x) first  Then use that result to evaluate f(x)

4. Numerically  f(g(1))g(1) = 5  f(5)  f(5) = 47  g(f(1)) = -3

5. Algebraically  Put the inner function into the outer function

6. Numerically

7. Graphically: Find (g o f)(-1) Find (f o g)(2)

8. Recognizing Composition of functions  Hint: Always look for the parenthesis

9. Question  What is f(undefined)?  Undefined so not included in the domain

10. To Determine the domain of Composite Functions  1. First, find the values to exclude from the inner function  2. Compose the 2 functions  Find the values excluded from the result  The domain of the composite function is the set of values that excludes both.

11. Jump ahead to Problem #9 on worksheet

12. 5.2 Inverse Functions  Verbally: one function undoes the other function. Returns the value back to whatever you started with, x.  Algebraically: “Show that 2 functions are inverses”  (f o g)(x) =x  (g o f)(x) = x  Numerically  f(x) = y (x,y)  f -1 (y)=x(y,x)

13. You try it…

14. 5.2 Inverse Functions  Graphically  Reflected over the line y=x  To find the graph of the inverse Function  Plot key points by switching the x and y values  Transform (x,y)  (y,x)

15. Worksheet #5

16. Worksheet #5 continued

17. Functions and Inverse Functions  Review: How do we know if a relation is a function  Vertical line test  Each x has only one y  Inverse Functions  Reflected over y=x  The original function must pass the horizontal line test  The inverse function (reflected) must pass the vertical line test  A function whose inverse is also a function is said to be:  One-to-one  “Strictly Monotonic”  Increasing (or decreasing) over the entire domain

18. Worksheet #6

19. What’s rwong with this…?

20. How to find inverse Functions  1. write using x and y notation  Still the same function  2. Switch the x and y variables  This is no longer the same function  (don’t use equals signs)  This is now the inverse function  3. Solve for y  Don’t forget to change back to inverse function notation

21. Try these

22. Help for the trickier problems  Write using x,y  Switch x and y  Move all the y terms to left side  Move other terms to right side  Factor out the y  Divide to solve for y

23. Start your Homework