1. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.2b Inverse Fcns

2. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.2 → Composite Functions  Any QUESTIONS About HomeWork §9.2 → HW-43 9.2 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 3 Bruce Mayer, PE Chabot College Mathematics Inverse & One-to-One Functions  Let’s view the following two functions as relations, or correspondences: Maine 1 Illinois 7 Iowa 2 Ohio 3 Domain Range (inputs) (outputs) States ball Ann rope Jim phone Jack car Domain Range (inputs) (outputs) Toys

4. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 4 Bruce Mayer, PE Chabot College Mathematics Inverse & One-to-One Functions  Suppose we reverse the arrows. We obtain what is called the inverse relation. Are these inverse relations functions? Maine 1 Illinois 7 Iowa 2 Ohio 3 Range Domain (inputs) (outputs) States ball Ann rope Jim phone Jack Car Range Domain (inputs) (outputs) Toys

5. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 5 Bruce Mayer, PE Chabot College Mathematics Inverse & One-to-One Functions Maine 1 Illinois 7 Iowa 2 Ohio 3 Range Domain (inputs) (outputs) States ball Ann rope Jim phone Jack Car Range Domain (inputs) (outputs) Toys  Recall that for each input, a function provides exactly one output. The inverse of “States” correspondence IS a function, but the inverse of “Toys” is NOT.

6. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 6 Bruce Mayer, PE Chabot College Mathematics One-to-One for “States” Fcn  In the States function, different inputs have different outputs, so it is a one-to-one function.  In the Toys function, rope and phone are both paired with Jim.  Thus the Toy function is NOT one-to-one.

7. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 7 Bruce Mayer, PE Chabot College Mathematics One-to-One Summarized  A function f is one-to-one if different inputs have different outputs. That is, if for a and b in the domain of f with a ≠ b we have f(a) ≠ f(b) then the function f is one-to-one.  If a function is one-to-one, then its INVERSE correspondence is ALSO a FUNCTION.

8. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 8 Bruce Mayer, PE Chabot College Mathematics One-to-One Fcn Graphically  Each y-value in the range corresponds to only one x-value in the domain i.e.; Each x has a Unique y

9. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 9 Bruce Mayer, PE Chabot College Mathematics NOT a One-to-One Fcn  The y-value y 2 in the range corresponds to TWO x-values, x 2 and x 3, in the domain.

10. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 10 Bruce Mayer, PE Chabot College Mathematics NOT a Function at All  The x-value x 2 in the domain corresponds to the TWO y-values, y 2 and y 3, in the range.

11. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 11 Bruce Mayer, PE Chabot College Mathematics Definition of Inverse Function  Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f −1.  If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f −1, and we write x = f −1 (y). We have y = f (x) if and only if f −1 (y) = x.

12. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  f-values ↔ f -1 -values  Assume that f is a one-to-one function. a.If f(3) = 5, find f -1 (5) b.If f -1 (−1) = 7, find f(7)  Solution: Recall that y = f(x) if and only if f -1 (y) = x a.Let x = 3 and y = 5. Now 5 = f(3) if and only if f −1 (5) = 3. Thus, f −1 (5) = 3. b.Let y = −1 and x = 7. Now, f −1 (−1) = 7 if and only if f(7) = −1. Thus, f (7) = −1.

13. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 13 Bruce Mayer, PE Chabot College Mathematics Inverse Function Property  Let f denote a one-to-one function. Then for every x in the domain of f –1. 1. for every x in the domain of f. 2.

14. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Inverse Fcn Property  Let f(x) = x 3 + 1. Show that  Soln:

15. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 15 Bruce Mayer, PE Chabot College Mathematics UNIQUE Inverse Fcn Property  Let f denote a one-to-one function. Then if g is any function such that g = f –1. That is, g is the inverse function of f. for every x in the domain of g and for every x in the domain of f, then

16. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 16 Bruce Mayer, PE Chabot College Mathematics Verify Inverse Functions  Verify that the following pairs of functions are inverses of each other:  Solution: From the composition of f & g.

17. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 17 Bruce Mayer, PE Chabot College Mathematics Verify Inverse Functions  Solution (cont.): Now Find  Observe:  This Verifies that f and g are indeed inverses of each other.

18. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Find Inverse of a Fcn  Given that f(x) = 5x − 2 is one-to-one, then find an equation for its inverse  Solution: f (x) = 5x – 2 y = 5x – 2 x = 5y – 2 Replace f(x) with y Interchange x and y Solve for y Replace y with f -1 (x)

19. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 19 Bruce Mayer, PE Chabot College Mathematics Procedure for finding f −1 1.Replace f(x) by y in the equation for f(x). 2.Interchange x and y. 3.Solve the equation in Step 2 for y. 4.Replace y with f −1 (x).

20. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Find the Inverse  Find the inverse of the one-to-one function  Solution: Step 1 Step 2 Step 3

21. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Find the Inverse Step 3 (cont.) Step 4

22. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Find Domain & Range  Find the Domain & Range of the function  Solution: Domain of f, all real numbers x such that x ≠ 2, in interval notation (−∞, 2)U(2, −∞)  Range of f is the domain of f −1  Range of f is (−∞, 1) U (1, −∞)

23. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 23 Bruce Mayer, PE Chabot College Mathematics Inverse Function Machine  Let’s consider inverses of functions in terms of function machines. Suppose that a one-to-one function f, has been programmed into a machine.  If the machine has a reverse switch, when the switch is thrown, the machine performs the inverse function, f −1. Inputs then enter at the opposite end, and the entire process is reversed.

24. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 24 Bruce Mayer, PE Chabot College Mathematics Reverse Switch Graphically Forward Reverse

25. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 25 Bruce Mayer, PE Chabot College Mathematics Horizontal Line Test  Recall that to be a Function an (x,y) relation must pass the VERTICAL LINE test  In order for a function to have an inverse that is a function, it must pass the HORIZONTAL-LINE test as well  NOT a Function – Fails the Vertical Line Test

26. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 26 Bruce Mayer, PE Chabot College Mathematics Horizontal Line Test Defined  If it is impossible to draw a horizontal line that intersects a function’s graph more than once, then the function is one-to-one.  For every one-to-one function, an inverse function exists.  A Function withOUT and Inverse – Fails the Horizontal Line Test (not 1-to-1)

27. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Horizontal Line Test  Determine whether the function f(x) = x 2 + 1 is one-to-one and thus has an inverse fcn.  The graph of f is shown. Many horizontal lines cross the graph more than once. For example, the line y = 2 crosses where the first coordinates are 1 and −1. Although they have different inputs, they have the same output: f(−1) = 2 = f(1). The function is NOT one-to-one, therefore NO inverse function exists x y -5 -4 -3 -2 -1 1 2 3 4 5 4 3 6 2 5 1 -2 7 8

28. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Horizontal Ln Test  Use the horizontal-line test to determine which of the following fcns are 1-to-1 a.b.  Soln a. No horizontal line intersects the graph of f in more than one point, therefore the function f is one-to-one

29. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Horizontal Ln Test  Use the horizontal-line test to determine which of the following fcns are 1-to-1 a.b.  Soln b. No horizontal line intersects the graph of f in more than one point, therefore the function f is 1-to-1

30. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 30 Bruce Mayer, PE Chabot College Mathematics Graphing Fcns and Their Inverses  How do the graphs of a function and its inverse compare?

31. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Graphs Inverse Fcn  Graph f(x) = 5x − 2 and f −1 (x) = (x + 2)/5 on the same set of axes and compare  Solution: f (x) = 5x – 2 f -1 (x) = (x + 2)/5  Note that the graph of f −1 (x) can be drawn by reflecting the graph of f across the line y = x.x.  When x and y are interchanged to find a formula for f −1 (x), we are, in effect, Reflecting or Flipping the graph of f.f.

32. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 32 Bruce Mayer, PE Chabot College Mathematics Visualizing Inverses  The graph of f −1 is a REFLECTION of the graph of f across the line y = x.

33. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Use y = x Mirror Ln  The graph of the function f is shown at Lower Right. Sketch the graph of the f −1  Soln

34. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Inverse or Not?  Ray’s Music Mart has six employees. The first table lists the first names and the Social Security numbers of the employees, and the second table lists the first names and the ages of the employees a.Find the inverse of the function defined by the first table, and determine whether the inverse relation is a function b.Find the inverse of the function defined by the second table, and determine whether the inverse relation is a function

35. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Inverse or Not? Dwayne590-56-4932 Sophia599-23-1746 Desmonde264-31-4958 Carl432-77-6602 Anna195-37-4165 Sal543-71-8026  Solution: Every y-value corresponds to exactly one x-value. Thus the inverse of the function defined in this table is a function

36. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Inverse or Not?  Solution: There is more than one x-value that corresponds to a y-value. For example, the age of 24 yields the names Dwayne and Anna. Thus the inverse of the function defined in this table is NOT a function. Dwayne24 Sophia26 Desmonde42 Carl51 Anna24 Sal26

37. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Hydrostatic Pressure  The formula for finding the water pressure p (in pounds per square inch, or psi), at a depth d (in feet) below the surface is  A pressure gauge on a Diving Bell breaks and shows a reading of 1800 psi. Determine how far below the surface the bell was when the gauge failed

38. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  HydroStatic P  Solution: The depth is given by the inverse of  Solve the Inverse Eqn for p Let p = 1800 psi  The Diving Bell was 3960 feet below the surface when the gauge failed

39. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 39 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.2 Exercise Set 38, 42, 60, 68, 76  Some Temperature Scales

40. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 40 Bruce Mayer, PE Chabot College Mathematics All Done for Today Old Style Diving Bell

41. BMayer@ChabotCollege.edu MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt 41 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –