Week 2 Rational Function, Equation and Inequality -Autosaved-.pptx

general math

1. Lesson 1 Representing Real-life Situations using Rational Function General Mathematics

2. A polynomial function p of degree n is a function that can be written in the form Where and n is a positive integer. Each summand is a term of the polynomial function. The constants are the coefficients . The leading coefficient is and the constant term is .  

3. Rational function a function of the form where p(x) and q(x) are polynomial functions, and q(x) is not the zero function, . The domain of f(x) is all values of x where .  

4. Example 1. An object is to travel a distance of 10 meters. Express the velocity v as a function of travel time t , in seconds   Solution. The following table of values show v for the various values of t. The function can represent v as a function of t.   t(seconds) 1 2 4 5 10 or 10 5 2.5 2 1 t(seconds) 1 2 4 5 10 10 5 2.5 2 1

5. Example 2. Suppose that (in mg/mL) represents the concentration of a drug in a patient’s bloodstream t hours after the drug was administered. Construct a table of values for for . Round off answers to three decimal places. Use the table to sketch a graph and interpret the results  

6. Solution : Using the rational function   We will substitute the values for   t 1 2 5 10 c(t) 2.5 2 0.962 0.495          

7. Solution: To get the graph of the function, we will use the table of values t 1 2 5 10 c(t) 2.5 2 0.9262 0.495

8. Interpretation The graph indicates that the maximum drug concentration occurs around 1 hour after the drug was administered. After 1 hour, the graph suggests that drug concentration decreases until it is almost zero.

9. Lesson 2 Rational Function, Equations And Inequalities General Mathematics

10. The table below shows the difference among ration equations, rational inequalities, and rational functions   Rational Equation Rational Inequality Rational Function Definition An equation involving rational expressions An inequality involving rational expressions A function of the form of where p(x) and q(x) is not the zero function Example   Rational Equation Rational Inequality Rational Function Definition An equation involving rational expressions An inequality involving rational expressions Example

11. A rational equation or inequality can be solved for all x values that satisfy the equation or inequality. Whereas we solve an equation or inequality, we do not “solve” functions. Rather, a function expresses a relationship between two variables (such as x and y), and can be represented by a table of values

12. Lesson 3 Solving Rational Equation and Inequalities General Mathematics

13. Procedure for Solving Rational Equations To solve rational equations: Eliminate denominators by multiplying each term of the equation by the least common denominator (LCD). Note that eliminating denominators may introduce extraneous solution. Check the solutions of the transformed equation with the original equation.

14. Solve for x.   Solution  

15. Solve for   Solution: The LCD is Upon reaching this step, we can use strategies for solving polynomial equations.  

16. Check for extraneous solutions by substituting the answers back into the original form. Since will make the original equation undefined, it is an extraneous solution. Since satisfies the original equation, it is the only solution.  

17. Example 3 : In an inter-barangay basketball league, the team from Barangay Parang has won 12 out of 25 games, a winning percentage of 48%. How many games should they win in a row to improve their win percentage to 60%

18. Solution. Let x represent the number of games that they need to win to raise their percentage to 60%. The team has already won 12 out of their 25 games. If they win x games in a row to increase their percentage to 60%, then they would have played games out of games. The equation is    

19. Solution: Since is the only denominator, we multiply it to both sides of the equation. We then solve the resulting equation:    

20. Since represents the number of games, this should be an integer. Therefore Barangay Parang needs to win 8 games in a row to raise their winning percentage to 60%  

21. Procedure for Solving Rational Inequalities To solve rational inequalities: Rewrite the inequality as a single fraction on one side of the inequality symbol and 0 on the other side

22. Determine over what intervals the fraction takes on positive and negative values. Locate the for which the rational expression is zero or undefined (factoring the numerator and denominator is a useful strategy) Mark the numbers found in (i) on a number line. Use a shaded circle to indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers partition the number line into intervals Select a test point within the interior of each interval in (ii). The sign of the rational expression at this test point is also the sign of the rational expression at each interior point in the aforementioned interval Summarize the intervals containing the solutions  

23. Warning! It is not valid to multiply both sides of an inequality by a variable. Recall that Multiplying both sides of an inequality by a positive number retains the direction of the inequality; and Multiplying both sides of an inequality by a negative number reverses the direction of the inequality Since the sign of a variable is unknown, then it is not valid to multiply both sides of an inequality by a variable.

24. Interval and Set Notation An inequality may have infinitely many solutions. The set of all solutions can be expressed using set notation or interval notation. These notations are presented in the table below:

25. Example 4: Solve the inequality   Solution: (a) Rewrite the inequality as a single fraction on one side, and 0 on the other side.  

26. (b) The value is included in the solution since it makes the fraction equal to zero, while makes the fraction undefined. Mark these on a number line. Use a shaded circle for and an unshaded circle for  

27. (c) Choose convenient test points in the interval determined by -1 and 1 to determine the sign of in these intervals. Construct a table of signs as show below.  

28. (d) Since we are looking for the intervals where the fractions is positive or zero, we determine the solution intervals to be .   The solution set is .  

29. Example 5: Solve:   Solution. (a) Rewrite as an inequality with zero on one side.  

30. (b) The fraction will be zero for and undefined for 0 and 2. Plot on a number line. Use hollow circle since these values are not part of the solutions.  

31. c) Construct a table of signs to determine the sign of the function in each interval determined by -1, 0, and 2.

32. d) Summarize the intervals satisfying the inequality. The solution set of the inequality is the set  

33. Lesson 4 Representation of Rational Functions General Mathematics

34. Scenario 1: Average speed (or velocity) can be computed by the formula . Consider a 100-meter track used for foot races. The speed of a runner can be computed by taking the time it will take him to run the track and applying it to the formula , since the distance is fixed at 100 meters  

35. Example 1: Represent the speed of a runner as a function of the time it takes to run 100 meters in the track. Solution. Since the speed of a runner depends on the time it takes the runner to run 100 meters, we can represent speed as a function of time. Let x represent the time it takes the runner to run 100 meters. Then the speed can be represented as a function as follows:  

36. Example 2: Continuing the scenario 1 above, construct a table of values for the speed of a runner against different run times Solution. Let x be the run time and be the speed of the runner in meters per second, where The table of values for run times from 10 seconds to 20 seconds is as follows   10 12 14 16 18 20 10 8.33 7.14 6.25 5.56 5 10 12 14 16 18 20 10 8.33 7.14 6.25 5.56 5            

37. From the table we can observe that the speed decreases with time. We can use a graph to determine if the points on the function follow a smooth curve or a straight line.

38. Example 3: Plot the points on the table of values on a Cartesian plane. Determine if the points on the function follow a smooth curve or a straight line.   Solution. Assign points on the Cartesian Plane for each entry on the table of values above: Plot these points on the Cartesian plane.  

40. For the 100-meter dash scenario, we have constructed a function of a speed against time and represented our function with a table of values and a graph that follow a smooth curve.

41. Lesson 5 Domain and Range of Rational Function General Mathematics

42. Finding the Domain and Range The domain of a rational function is all the values of x that will not make equal to zero To find the range of a rational function is by finding the domain of the inverse function. Another way to find the range of a rational function is to find the value of horizontal asymptote.  

43. Example 1 : Find the domain and range of the rational function   Solution: (a) To find the domain we first equate the denominator to zero Therefore, the domain of is the set of all real numbers except 3.  

44. Solution (b): To find the range we need to the get the inverse function of    

45. We will now equate the denominator to zero to find the limit of range   Therefore, the range of is the set of all real numbers except zero.  

46. Example 2: Find the domain and range of the rational function   Solution: (a) to find the domain we will equate the denominator to zero Therefore, the domain of is the set of all real numbers except 3 and 1  

47. (b) To find the range we will apply the concept of horizontal asymptote Let n be the degree of the numerator and m be the degree of the denominator If , the horizontal asymptote is y=0 If , the horizontal asymptotes is , where a and b is the leading coefficient of the numerator and b is the leading coefficient of the denominator. If , there is no horizontal asymptote.  

48. Expand the function Both numerator and denominator have the same degree which is 2. Therefore, we will use the concept of horizontal asymptote and  

49. Therefore, the range of is the set of all real numbers except 1.  

50. Thank You!