1. Chapter 1 Functions and Their Graphs

2. Warm Up 1.6  A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at a constant rate of 0.03 inches per second and that r = 48 inches at time t = 0. Determine an equation that models the volume V of the balloon at time t and find the volume when t = 300 seconds. Note: Volume of a sphere. 2

3. 1.6Inverse Functions Objectives:  Find inverse functions numerically and algebraically.  Verify that two functions are inverse functions of each other.  Determine if functions are one-to-one.  Use graphs of functions to decide whether functions have inverse functions. 3

4. Consider This … A highway crew is painting the center line on a road. Knowing the crew’s previous work record, it is possible to predict how much of the stripe the crew will have painted at any time during a normal 8-hour shift. It may also be possible to tell how long the crew has been working by how much stripe has been painted. 4

5. Mathematically …  For the first case, the distance is dependent on time.  For the second, time is dependent on the distance. 5

6. Inverses  If a new relation is formed by interchanging the input and output variables in a given relation, the two relations are inverses of each other.  If both relations are functions, they are called inverse functions.  Notation: The inverse of f (x) is f -1 (x). 6

7. Inverse of a Function Numerically  Let d be the number of miles a highway crew paints in an 8-hour shift.  Let t be the number of hours the crew has been on the job.  Find f (1), f (2), etc.  Find g(1), g(2), etc.  Inputs & outputs are interchanged. 7 t (hours) d (miles) 10.2 20.6 31.0 41.4 51.8 62.2 72.6 83.0

8. Graphs of the Functions 8 d = f (t)t = g(d)

9. Inverse of a Function Graphically  When a function is graphed on the same set of axes as its inverse, we see that the function and its inverse are reflections across the line y = x. 9

10. Inverse of a Function Algebraically  To find the inverse of a function algebraically:  Exchange x -values and y -values.  Solve for y.  Call the original function f (x) and the inverse function f -1 (x). 10

11. Algebraic Inverse Example 11 Linear function for the function f. or Interchange variables. Solve for y in terms of x. Label function as f (x) and inverse as f -1 (x).

12. Composition of Inverse Functions  The composition of a function f with its inverse f -1 results in the identity function.  The functions f and f -1 “undo” one another.  That is, and. 12

13. Domain and Range of Inverse Functions  If we interchange the x -values and the y -values to create the inverse of a function, it must follow that:  The domain of f is the range of f -1 and  The range of f is the domain of f -1.  Likewise,  The domain of f -1 is the range of f and  The range of f -1 is the domain of f. 13

14. Existence of an Inverse  How do we know that a relation is a function?  It passes the Vertical Line Test.  Each input has only one output.  For the function to have an inverse,  Each output must have only one input.  The function must be one-to-one. 14

15. One-to-One Functions  A one-to-one function has exactly one y for each x and exactly one x for each y.  If a function is one-to-one, it will pass the Horizontal Line Test.  A function has an inverse if and only if it is one-to-one.  If f is increasing on its entire domain or if it is decreasing on its entire domain, then f is one-to-one and, therefore, has an inverse. 15

16. Example 1  Given. a. Make a table of values for f (–2), f (–1), f (0), f (1), and f (2). From the numbers in the table, explain why you cannot find the value of x if f (x) = 2.5. How does this result indicate that we cannot find an inverse for f ? b. Plot the five points for f and connect with a smooth curve. Do the same for the inverse relation (on the same set of axes). How does the graph confirm that the inverse function for f cannot be found? c. Plot the line y = x on the same set of axes. What do you see? 16

17. Homework 1.6  Worksheet 1.6  # 6, 7, 9, 11, 17, 20, 21 – 24 (matching), 27, 32, 34, 39, 56, 57, 61, 62, 67, 68, 83, 84 17