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Chapt 6. Rational Expressions, Functions, and Equations

Chapt 6. Rational Expressions, Functions, and Equations. 6.1 Rational Expressions and Functions. Rational Expression Polynomial divided by non-zero polynomial 120x / (100 – x) (3x 2 - 12xy – 15y 2 ) / (6x 3 – 6xy 2 ) Rational Function Function defined by a rational expression

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Chapt 6. Rational Expressions, Functions, and Equations

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  1. Chapt 6. Rational Expressions, Functions, and Equations

  2. 6.1 Rational Expressions and Functions • Rational Expression • Polynomial divided by non-zero polynomial • 120x / (100 – x)(3x2 - 12xy – 15y2) / (6x3 – 6xy2) • Rational Function • Function defined by a rational expression • f(x) = (120x) / (100 – x)

  3. Evaluating a Function • Given: f(x) = 120x / (100 – x) • Evaluate: f(20)f(20) = 120(20) / (100 – (20)) = 2400 / 80 = 30f(40) = 120(40) / (100 – (40)) = 4800 / 60 = 80

  4. Domain of a Rational Function • Given: The cost (in $1000) of cleaning up a polluted lake is a function of the percentage (x) of the lake’s pollutants to be removed. It is given by the following function.f(x) = 120x / (100 – x) • What is the cost of cleaning up 50% of the pollutants? • f(50) = 120(50) / (100 – 50) = 120

  5. Domain of a Rational Function • Given the last function: f(x) = 120x / (100 – x) • What are the possible values of x? • Answer: • x ≠ 100 • x cannot be negative (in practical cases) • Domain of f: • [0, 2) U (2, 100]

  6. Domain of a Rational Function • Given: f(x) = (2x + 1) / (2x2 – x – 1) • What is the domain of f? • Solution:(2x2 – x – 1)(2x + 1)(x – 1) = 02x + 1 = 0 x – 1 = 0x = -1/2 x = 1Domain of f: (-∞ , -1/2) U (-1/2, 1) U (1, ∞) -1/2 1

  7. Your Turn • Given: f(x) = (x – 5) / (2x2 + 5x – 3) • Find the domain of f. • Solution: 2x2 + 5x – 3(2x - 1)(x + 3) = 02x – 1 = 0 x + 3 = 0x = ½ x = -3 • Domain of f:(-∞ , -3) U (-3, 1/2) U (1/2, ∞)

  8. Simplifying Rational Expressions • Simplify: (x2 + 4x + 3) / (x + 1) x2 + 4x + 3 (x + 1)(x + 3)--------------- = ------------------ = x + 1, x ≠ 1 x + 1 (x + 1) y = x + 1 y = (x2 + 4x + 3)/(x + 1)

  9. Your Turn • Simplify • (x2 + 7x + 10) / (x + 2) • = (x + 2)(x + 5) / (x + 2)= x + 5, x ≠ -2 • (x2 – 7x – 18) / (2x2 + 3x – 2) • = (x + 2)(x – 9) / (2x - 1)(x + 2)= (x – 9) / (2x – 1), x ≠ -2 and x ≠ 1/2

  10. Multiplying Rational Expressions • Multiply • x + 4 x2 – 4x - 21 -------- ∙ ---------------- x – 7 x2 – 16 • x + 4 (x – 7)(x + 3)= -------- · ------------------- x – 7 (x – 4)(x + 4) • x + 3= -------- x – 4

  11. Dividing Rational Expressions • Divide (y2 – 25) / (2y – 2) (y2 + 10y +25) / (y2 + 4y – 5) • = (y2 – 25) / (2y – 2) ∙ (y2 + 4y – 5)/(y2 + 10y + 25) • (y – 5)(y + 5) (y + 5)(y – 1) = ------------------ ∙ ------------------- 2(y – 1) (y + 5)(y + 5) y - 5 • = -------- 2

  12. Your Turn • Simplify the following • x2 + xy 4x – 4y ----------- · ----------x2 – y2 x • x(x + y) 4(x – y) = ------------------ · ------------ (x – y)(x + y) x • = 4

  13. Your Turn • Simplify • (y2 + y) / (y2 – 4) (y2 + 5y + 6) / (y2 – 1) • = (y2 + y) / (y2 – 4) ∙ (y2 – 1) / (y2 + 5y + 6) • y(y + 1) (y + 2)(y + 3)= ----------------- ∙ ------------------ (y – 2)(y + 2) (y - 1)(y + 1) • y(y + 3)= ------------------- (y – 2)(y – 1)

  14. 6.2 Adding and Subtracting Rational Expressions • Add • x2 + 2x – 2 x + 12 ------------------- + ------------------ x2 + 3x – 10 x2 + 3x – 10 • x2 + 2x – 2 + x + 12 (x + 2) (x + 5) = ---------------------------- = -------------------- x2 + 3x – 10 (x + 5)(x – 2) (x + 2) (x + 5) (x + 2)= -------------------- = ------------- (x + 5)(x – 2) (x – 2)

  15. Your Turn • Add • x2 – 5x – 15 2x + 5 ------------------- + ------------------ x2 + 5x + 6 x2 + 5x + 6 • Solution x2 – 5x – 15 + 2x + 5 (x - 5) (x + 3) = ------------------------------ = -------------------- x2 + 5x + 6 (x + 3)(x + 2) (x + 2) (x + 5) (x + 2) = -------------------- = ------------- (x + 5)(x – 2) (x – 2)

  16. Your Turn • Subtract • 3y3 – 5x3 4y3 – 6x3 --------------- - --------------- x2 – y2 x2 – y2 • Solution 3y3 – 5x3 - (4y3 – 6x3) 3y3 – 5x3 - 4y3 + 6x3 = ------------------------------- = ---------------------------- x2 – y2x2 – y2 x3 - y3 (x – y)(x2 + xy + y2) (x2 + xy + y2) = ---------------- = --------------------------- = -------------------- x2 – y2 (x – y)(x + y) (x + y)

  17. Finding the Least Common Denominator • Find the LCD of: 7/6x2 & 2/9x • Solution: • Factor denominators6x2 2, 3, x, x9x  3, 3, x • List all factors of 1st Denominator—2, 3, x, x • Add factors of 2nd dominator not in the list—2, 3, x, x, & 3 • LCD: product of all factors in the list—18x2

  18. Finding the Least Common Denominator • Find the LCD of: 7/(5x2 + 15x) and 9/(x2 + 6x + 9) • Solution: • Find factors in 1st denominator5x2 + 15x  5x(x + 3) • Find factors of 2nd denominatorx2 + 6x + 9  (x + 3)(x + 3) • List factors of 1st denominator5x(x + 3) • Include in the list those factors in 2nd denominator not found in 1st5x(x + 3)(x + 3) or 5x(x + 3)2

  19. Your Turn • Find the LCD of: • 7 / (y2 – 4) and 15 / (y2 + 2y) • 1st den: y2 – 4 = (y + 2)(y – 2) • 2nd den: y2 + 2y = y(y + 2) • LCD: (y + 2)(y – 2)y • 3/(y2 – 5y – 6) and 6/(y2 – 4y – 5) • 1st den: y2 – 5y – 6 = (y – 6)(y + 1) • 2nd den: y2 – 4y – 5 = (y – 5)(y + 1) • LCD: (y – 6)(y + 1)(y – 5)

  20. 6.3 Complex Rational Expressions • Given: • p =principal (amount borrowed) • r = monthly interest rate • n = number of monthly payments • A = amount of month payment • prA = ----------------------- 1 1 - -------------- (1 + r)n • ComplexRation Expression – has complex rational expression in numerator or denominator

  21. Simplifying Complex Rational Expression • Simplify: 1 y--- + --- x x2----------- 1 x--- + --- y y2 • Find the LCD: x x y y = x2y2 • Multiply all terms by x2y2 / x2y2 = 1

  22. (x2y2)1 (x2y2)y xy2 + y3 ---------- + ----------- ---------------- (x2y2)x (x2y2)x2 x2y2 ----------------------------- = --------------------- (x2y2)1 (x2y2)x x2y + x3---------- + ----------- ---------------- (x2y2)y (x2y2)y2 x2y2 xy2 + y3 y2(x + y) y2------------- = -------------- = -----x2y + x3 x2(y + x) x2

  23. YourTurn • ((x/y) – 1) / ((x2/y2) – 1)) • Hint: What is the LCD? • Solution: (xy – y2) / (x2 – y2) = y / (x + y) • (1/(x + h) – 1/x) / h • Hint: What is the LCD? • Solution: -1/(x( + h))

  24. Skip • 6.4 Division of Polynomial Expressions • 6.5 Synthetic Division

  25. 6.6 Rational Equations • Given: • Cost (in $1000) of cleaning a lake 120xf(x) = ---------- 100 – xwhere x = % of pollutants to be eliminated • Question: • If $80,000 is appropriated for the cleanup, what % of pollutants can be eliminated?

  26. 120xf(x) = ----------- 100 – x • Solution: • 200x80 = ----------- 100 – x80(100 – x) = 200x8000 – 80x = 200x8000 = 280xx = 25.7(%)

  27. Solving Rational Equation • Solve: x + 6 x + 24 -------- + ---------- = 2 2x 5x • Note: x ≠ 0x + 6 x + 24 10x -------- + ---------- = 10x 2 2x 5x5(x + 6) + 2(x + 24) = 20x5x + 30 + 2x + 48 = 20x78 = 13xx = 6

  28. Check • Solve: x + 6 x + 24 -------- + ---------- = 2 2x 5x • Note: x ≠ 06 + 6 6 + 24 ?------- + ---------- = 22(6) 5(6) 12 30------- + ------- = 2 12 30

  29. Solving Rational Equation (2) • Solve: x 3 -------- = ---------- + 9 x – 3 x – 3 • Note: x ≠ 3x 3 (x – 3) -------- = (x – 3) --------- + 9 x - 3 x - 3x = 3 + (x – 3)9 x = 3 + 9x – 27x = -24 + 9x24 = 8xx = 3But x cannot be 3. Thus, no solution.

  30. Solving Rational Equation (3) • Solve x 9---- + ----- = 4 3 x • Note: x ≠ 0x 9 (3x) ----- + ---- = (3x) 4 3 x x(x) + 3(9) = 12x x2 + 27 = 12xx2 – 12x + 27 = 0(x – 3)(x – 9) = 0x = 3, x = 9

  31. Check • Solve x 9---- + ----- = 4 x = 3, x = 9 3 x • Note: x ≠ 0 3 9 ? 9 9 ?--- + --- = 4 ---- + ---- = 4 3 3 3 9 1 + 3 = 4 3 + 1 = 4

  32. Your Turn • Solve: • x + 4 x + 20-------- + ---------- = 3 2x 3x • Solution: • x ≠ 0 x + 4 x + 206x -------- + ---------- = 6x 3 2x 3x3(x + 4) + 2(x + 20) = 18x3x + 12 + 2x + 40 = 18x52 = 13xx = 4

  33. Your Turn • Solve: • 2x 6 -28-------- + --------- = ------------ x – 3 x + 3 x2 - 9 • Solution: • x ≠ 3, x ≠ -3 2x 6 -28 (x – 3)(x + 3) ---------- + ---------- = (x – 3)(x + 3) ----------- (x – 3) (x + 3) x2 - 9(x + 3)2x + (x – 3)6 = -282x2 + 6x + 6x – 18 = -282x2 + 12x + 10 = 0(2x + 2)(x + 5) = 0x = -1, x = -5

  34. 6.7 Applications • Suppose: Tom can complete a Web site in 15 hours, while her friend Amy can complete it in 10 hours. Working together, how many hours will it take to complete one job? • Solution: • Hours working together: x • Hour with Tom alone: 15 • Hours with Amy alone: 10 • Tom’s rate: 1/15 per hour • Amy’s rate: 1/10 per hour

  35. Find an equation • Rate x Time = 1 job • 1 1 x ---- + ---- = 1 15 10(30) x (30) x (30) 1------ ---- + ------ ---- = ------(30) 15 (30) 10 (30)2x + 3x = 305x = 30 • x = 6 (hours)

  36. Application • You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. If the round trip takes 2 hours, what is your average rate on the outgoing trip to work

  37. Solution • Average rate outgoing (mph): x • Average rate returning: x + 30 • Find Equation • distance = rate x time • time = distance / rate • (time going) + (time returning) = 2 • 40/x + 40/(x + 30) = 2 • (x + 30)40 + 40x = 2x(x + 30) • 40x + 1200 + 40x = 2x2 + 60x • 0 = 2x2 - 20x – 1200 • 0 = x2 - 10x – 600 • 0 = (x – 30)(x + 20) • x = 30; • x = -20 (has no interpretation)

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