1. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.1 Rational Fcn Mult & Div

2. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.7 → PolyNomial Equation Applications  Any QUESTIONS About HomeWork §5.7 → HW-17 5.7 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 3 Bruce Mayer, PE Chabot College Mathematics Recall Rational Function  A rational function is a function, f(x), that is a quotient of two polynomials; i.e.   Where where p(x) and q(x) are polynomials and where q(x) is NOT the ZERO polynomial. The domain of f consists of all inputs x for which q(x) ≠ 0.

4. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 4 Bruce Mayer, PE Chabot College Mathematics Rational FUNCTION Example  RATIONAL FUNCTION ≡ a function expressed in terms of rational expressions  Example  Find f(3) for this Rational Function:  SOLUTION

5. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 5 Bruce Mayer, PE Chabot College Mathematics Find the Domain of a Rational Fcn 1.Write an equation that sets the denominator of the rational function equal to 0. 2.Solve the denominator equation. 3.Exclude the value(s) found in step 2 from the function’s domain.

6. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Find Domain  Find the Domain for  SOLUTION Set the denominator equal to 0. Factor out the monomial GCF, y. Use the zero products theorem.  The fcn is undefined for y = 0, −4, or −1, so the domain is {y|y  −4, −1, 0}. FOIL Factor the 2 nd Degree polynomial

7. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Find Domain  Find the Domain for  SOLUTION Find the values of x for which the denominator x 2 – 6x + 8 = 0, then exclude those values from the domain.  The fcn is undefined for x = 2, or 4, so the domain is {x|x  2, 4}. Interval Notation: (−∞,2)U(2,4)U(4,∞)

8. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION: x  1, so the graph has a vertical asymptote at x = 1. Find ordered pairs around the asymptote and then graph. x 44 22 00.5 y  3/5 11 33 66 x1.5245 y6313/4

9. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 9 Bruce Mayer, PE Chabot College Mathematics Simplifying Rational Expressions and Functions  As in arithmetic, rational expressions are simplified by “removing”, or “Dividing Out”, a factor equal to 1.  example removed the factor that equals 1 equals 1

10. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 10 Bruce Mayer, PE Chabot College Mathematics Maintain Domain  Because rational expressions often appear when we are writing functions, it is important that the function’s domain not be changed as a result of simplifying. For example, the Domain of the function given by is assumed to be all real numbers for which the denominator is NONzero

11. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 11 Bruce Mayer, PE Chabot College Mathematics Maintain Domain  Thus for Rational Fcn:  In the previous example, we wrote F(x) in simplified form as  There is a serious problem with stating that these are equivalent; The Domains are NOT the same

12. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 12 Bruce Mayer, PE Chabot College Mathematics Maintain Domain  Why  The domain of the function given by  Thus the domain of G includes 5, but the domain of F does not. This problem can be addressed by specifying ≠

13. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Maintain Domain  Write this Fcn in Simplified form  SOLUTION: first factor the numerator and denominator, looking for the largest factor common to both. Once the greatest common factor is found, use it to write 1 and simplify as shown on the next slide

14. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Maintain Domain Note that the domain of g = {x | x  2/3 and x  −7} by Factoring (see next) Factoring. The greatest common factor is (3x − 2). Rewriting as a product of two rational expressions. For x  2/3, we have (3x − 2)/(3x − 2) = 1.

15. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Maintain Domain  Thus the simplified form of Removing the factor 1. To keep the same domain, we specify that x  2/3.

16. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 16 Bruce Mayer, PE Chabot College Mathematics “Canceling” Confusion  The operation of Canceling is a ShortHand for DIVISION between Multiplication Chains  Canceling can ONLY be done when we have PURE MULTIPLICATION CHAINS both ABOVE & BELOW the Division Bar`

17. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 17 Bruce Mayer, PE Chabot College Mathematics Canceling Caveat  “Canceling” is a shortcut often used for removing a factor equal to 1 when working with fractions. Canceling removes multiplying factors equal to 1 in products. It cannot be done in sums or when adding expressions together. Simplifying the expression from the previous example might have been done faster as follows: When a factor that equals 1 is found, it is “canceled” as shown. Removing a factor equal to 1.

18. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 18 Bruce Mayer, PE Chabot College Mathematics Canceling Caveat Caution! Canceling is often performed incorrectly: Incorrect! Incorrect! Incorrect! In each situation, the expressions canceled are not both factors. Factors are parts of products. For example, 5 is not a factor of the numerator 5x – 2. If you can’t factor, you can’t cancel! When in doubt, do NOT cancel! To check that these are not equivalent, substitute a number for x.

19. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 19 Bruce Mayer, PE Chabot College Mathematics Simplifying Rational Expressions 1.Write the numerator and denominator in factored form. 2.Divide out all the common factors in the numerator and denominator; i.e., remove factors equal to ONE 3.Multiply the remaining factors in the numerator and the remaining factors in the denominator.

20. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Simplify  SOLUTION: Factor out the GCF. Factor the polynomial factors. Divide out common factors.

21. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 21 Bruce Mayer, PE Chabot College Mathematics Multiply Rational Expressions  The Product of Two Rational Expressions  To multiply rational expressions, multiply numerators and multiply denominators:  Then factor and simplify the result if possible.

22. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Multiplication  Multiply and, if possible, simplify. a)b)  SOLUTION a) MULTIPLICATION Chains → Canceling OK

23. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Multiplication  SOLUTION b) MULTIPLICATION Chains → Canceling OK

24. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Multiply & Simplify  Multiply and, if possible, simplify.  SOLUTION MULTIPLICATION Chains → Canceling OK

25. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 25 Bruce Mayer, PE Chabot College Mathematics Divide Rational Expressions  The Quotient of Two Rational Expressions  To divide by a rational expression, multiply by its reciprocal  Then factor and, if possible, simplify.

26. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Division  Divide and, if possible, simplify. a)b)  SOLUTION Multiplying by the reciprocal of the divisor Multiplying rational expressions a) b)

27. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Division  Divide and, if possible, simplify.  SOLN MULTIPLICATION Chains → Canceling OK

28. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Division  Divide and, if possible, simplify.  SOLUTION  MULTIPLICATION Chains → Canceling OK

29. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Division  Divide and, if possible, simplify.  SOLUTION  MULTIPLICATION Chains → Canceling OK

30. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Manufacturing Engr  The function given by  gives the time, in hours, for two machines, working together to complete a job that the 1 st machine could do alone in t hours and the 2 nd machine could do in 3t − 2 hours. How long will the two machines, working together, require for the job if the first machine alone would take (a) 2 hours? (b) 5 hours?

31. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Manufacturing Engr  SOLUTION (a) (b)

32. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 32 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.1 Exercise Set 22 (ppt), 26 (ppt), 114, 16, 46, 66, 86  More Rational Division  Since we are dividing fractions, we multiply by the reciprocal  Now, we follow the rule for multiplication  Factor and then cancel  Don't leave the numerator empty - put a one to hold the place.

33. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 33 Bruce Mayer, PE Chabot College Mathematics P6.1-22  Rational fcn Graph  Describe end-behavior of Graph at Far-Right  ANS: As x becomes large y = f(x) approaches, but never reaches, a value of 3

34. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 34 Bruce Mayer, PE Chabot College Mathematics P6.1-22  Rational fcn Graph  What is the Eqn for the Horizontal Asymptote:  ANS: y = f(x) approaches, but never reaches, a value of 3, to the Asymptote eqn y = 3

35. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 35 Bruce Mayer, PE Chabot College Mathematics P6.1-26  Rational fcn Graph  List 2 real No.s that are NOT function values of f  ANS: y = f(x) does not have a graph between y > 0 and y < 3. Thus two values for which there is NO f(x): y = 1 or y = 2

36. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 36 Bruce Mayer, PE Chabot College Mathematics P6.1-114  Smoking Diseases  Find P(9). Describe Meaning. ID pt on Graph  ANS: An incidence ratio of 9 indicates that 88.9% of Lung Cancer deaths are associated with Cigarette smoking

37. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 37 Bruce Mayer, PE Chabot College Mathematics P6.1-114  Smoking Diseases  ID P(9) on Graph

38. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 38 Bruce Mayer, PE Chabot College Mathematics All Done for Today Diophantus of Alexandria  The FIRST Algebraist In the 3 rd century, the Greek mathematician Diophantus of Alexandria wrote his book Arithmetica. Of the 13 parts originally written, only six still survive, but they provide the earliest record of an attempt to use symbols to represent unknown quantities.

39. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 39 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

40. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 40 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

41. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 41 Bruce Mayer, PE Chabot College Mathematics