1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 6.1 Rational Expressions and Functions: Multiplying and Dividing Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3. 3 Rational Expressions A rational expression consists of a polynomial divided by a nonzero polynomial (denominator cannot be equal to 0). A rational function is a function defined by a formula that is a rational expression. For example, the following is a rational function:

4. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 Rational ExpressionsEXAMPLE The rational function models the cost, f (x) in millions of dollars, to inoculate x% of the population against a particular strain of flu. The graph of the rational function is shown. Use the function’s equation to solve the following problem. Find and interpret f (60). Identify your solution as a point on the graph.

5. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 Rational ExpressionsCONTINUED

6. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 Rational ExpressionsSOLUTION We use substitution to evaluate a rational function, just as we did to evaluate other functions. CONTINUED This is the given rational function. Replace each occurrence of x with 60. Perform the indicated operations.

7. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 Rational Expressions Thus, f (60) = 195. This means that the cost to inoculate 60% of the population against a particular strain of the flu is $195 million. The figure below illustrates the solution by the point (60,195) on the graph of the rational function. CONTINUED (60,195)

8. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 Objective #1: Example

9. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 Objective #1: Example

10. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10

11. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Rational Expressions - DomainEXAMPLE Find the domain of f if The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x. SOLUTION Set the denominator equal to 0. Factor.

12. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 Rational Expressions - Domain Because 4 and 9 make the denominator zero, these are the values to exclude. Thus, Set each factor equal to 0. Solve the resulting equations. CONTINUED or

13. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Rational Expressions - DomainCONTINUED In this example, we excluded 4 and 9 from the domain. Unlike the graph of a polynomial which is continuous, this graph has two breaks in it – one at each of the excluded values. Since x cannot be 4 or 9, there is not a function value corresponding to either of those x values. At 4 and at 9, there will be dashed vertical lines called vertical asymptotes. The graph of the function will approach these vertical lines on each side as the x values draw closer and closer to each of them, but will not touch (cross) the vertical lines. The lines x = 4 and x = 9 each represent vertical asymptotes for this particular function.

14. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Objective #2: Example

15. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15 Objective #2: Example

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17. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Rational Expressions Asymptotes Vertical Asymptotes A vertical line that the graph of a function approaches, but does not touch. Horizontal Asymptotes A horizontal line that the graph of a function approaches as x gets very large or very small. The graph of a function may touch/cross its horizontal asymptote.

18. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Objective #3: Example

19. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 Objective #3: Example

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21. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21 Rational Expressions Simplifying Rational Expressions 1) Factor the numerator and the denominator completely. 2) Divide both the numerator and the denominator by any common factors.

22. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22 Simplifying Rational ExpressionsEXAMPLE Simplify: SOLUTION Factor the numerator and denominator. Divide out the common factor, x + 1. Simplify.

23. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23 Objective #4: Example

24. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Objective #4: Example

25. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Objective #4: Example

26. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26 Objective #4: Example

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28. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 Multiplying Rational Expressions 1) Factor all numerators and denominators completely. 2) Divide numerators and denominators by common factors. 3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators. Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator The quotient of two polynomials that have opposite signs and are additive inverses is –1.

29. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 29 Multiplying Rational ExpressionsEXAMPLE Multiply: SOLUTION Factor the numerators and denominators completely. Divide numerators and denominators by common factors. This is the original expression.

30. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Multiplying Rational Expressions Multiply the remaining factors in the numerators and in the denominators. CONTINUED

31. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Multiplying Rational ExpressionsEXAMPLE Multiply: SOLUTION Factor the numerators and denominators completely. This is the original expression. (-1) Divide numerators and denominators by common factors. Because 3 – y and y – 3 are opposite, their quotient is -1.

32. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Multiplying Rational ExpressionsCONTINUED or

33. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Objective #5: Example

34. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Objective #5: Example

35. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35 Objective #5: Example

36. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Objective #5: Example

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38. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 38 Dividing Rational Expressions If P, Q, R, and S are polynomials, where then Change division to multiplication. Replace with its reciprocal by interchanging its numerator and denominator.

39. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 39 Dividing Rational ExpressionsEXAMPLE Divide: SOLUTION Invert the divisor and multiply. Factor. This is the original expression.

40. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 40 Dividing Rational Expressions Divide numerators and denominators by common factors. CONTINUED Multiply the remaining factors in the numerators and in the denominators.

41. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 41 Objective #6: Example

42. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 42 Objective #6: Example