f g (0) g Link to prerequisite algebra material (For help with complex fractions and radicals.)

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1 88 Relations and Functions.5. Exercises To see all o the help resources associated with this section, click OSttS Chapter b. In Exercises - 0, use the pair o unctions and to ind the ollowin values i they exist For help with these exercises, click on one o the resources below: Addin and subtractin unctions. Multiplyin and dividin unctions. Link to prerequisite alebra material For help with complex ractions and radicals.. x = 3x + and x = 4 x 2. x = x 2 and x = 2x + 3. x = x 2 x and x = 2 x 2 4. x = 2x 3 and x = x 2 2x 3 5. x = x + 3 and x = 2x 6. x = 4 x and x = x x = 2x and x = 2x + 8. x = x 2 and x = 3 2x 3 9. x = x 2 and x = x 2 0. x = x 2 + and x = x 2 + In Exercises - 20, use the pair o unctions and to ind the domain o the indicated unction then ind and simpliy an expression or it. + x x x x For help with these exercises, click on one o the resources below: Addin and subtractin unctions. Multiplyin and dividin unctions. Link to prerequisite alebra material For help with complex ractions and radicals.. x = 2x + and x = x 2 2. x = 4x and x = 2x

2 .5 Function Arithmetic x = x 2 and x = 3x 4. x = x 2 x and x = 7x 5. x = x 2 4 and x = 3x x = x 2 + x + 6 and x = x x = x 2 and x = 2 x 8. x = x and x = x 9. x = x and x = x x = x 5 and x = x = x 5 In Exercises 2-45, ind and simpliy the dierence quotient For help with these exercises, click on the resource below: Simpliyin dierence quotients x + h x h or the iven unction. Link to prerequisite alebra material For help with complex ractions and radicals. 2. x = 2x x = 3x x = x = 3x 2 x 25. x = x 2 + 2x 26. x = 4x x = x x x = x x = mx + b where m x = ax 2 + bx + c where a 0 3. x = 2 x 32. x = 3 x 33. x = x x = 2 x x = 4x x = 3x x x = x x x = x2 2x x = x x = 2x + 4. x = 4x x = 4 x 43. x = ax + b, where a x = x x 45. x = 3 x. HINT: a b a 2 + ab + b 2 = a 3 b 3

3 90 Relations and Functions In Exercises 46-50, Cx denotes the cost to produce x items and px denotes the price-demand unction in the iven economic scenario. In each Exercise, do the ollowin: Find and interpret C0. Find and interpret p5 Find and simpliy P x. Find and interpret C0. Find and simpliy Rx. Solve P x = 0 and interpret. 46. The cost, in dollars, to produce x I d rather be a Sasquatch T-Shirts is Cx = 2x + 26, x 0 and the price-demand unction, in dollars per shirt, is px = 30 2x, 0 x The cost, in dollars, to produce x bottles o 00% All-Natural Certiied Free-Trade Oranic Sasquatch Tonic is Cx = 0x + 00, x 0 and the price-demand unction, in dollars per bottle, is px = 35 x, 0 x The cost, in cents, to produce x cups o Mountain Thunder Lemonade at Junior s Lemonade Stand is Cx = 8x + 240, x 0 and the price-demand unction, in cents per cup, is px = 90 3x, 0 x The daily cost, in dollars, to produce x Sasquatch Berry Pies Cx = 3x + 36, x 0 and the price-demand unction, in dollars per pie, is px = 2 0.5x, 0 x The monthly cost, in hundreds o dollars, to produce x custom built electric scooters is Cx = 20x + 000, x 0 and the price-demand unction, in hundreds o dollars per scooter, is px = 40 2x, 0 x 70. In Exercises 5-62, let be the unction deined by and let be the unction deined by = { 3, 4, 2, 2,, 0, 0,,, 3, 2, 4, 3, } = { 3, 2, 2, 0,, 4, 0, 0,, 3, 2,, 3, 2} Compute the indicated value i it exists

4 .5 Function Arithmetic 9. Let x = 2x + 6 and x = x 2 9. Checkpoint Quiz.5 Find the domain o the ollowin unctions and simpliy their expressions. a x b x 2. Let x = x 2 + 2x 3. Find and simpliy the dierence quotient, For worked out solutions to this quiz, click the link below: x + h x. h Quiz Solution

5 92 Relations and Functions.5.2 Answers. For x = 3x + and x = 4 x + 2 = 9 = 7 = 2 = = 4 2 = For x = x 2 and x = 2x = = 2 = 2 2 = 0 0 = 0 2 = For x = x 2 x and x = 2 x = 0 = 9 = 2 = = 0 2 = For x = 2x 3 and x = x 2 2x = 5 = 0 = 8 2 = = 0 2 = For x = x + 3 and x = 2x + 2 = = = 2 = 0 0 = 3 2 = 5 6. For x = 4 x and x = x = = + 5 = 0 2 = = 2 2 = 0

6 .5 Function Arithmetic For x = 2x and x = 2x+ + 2 = 2 5 = = = 2 0 = 0 2 = 2 8. For x = x 2 and x = 3 2x = 7 = 8 5 = 4 2 = = 0 2 = For x = x 2 and x = x = 7 4 = 0 = 0 2 = 0 is undeined. 2 = 6 0. For x = x 2 + and x = x = 26 5 = 3 2 = = 0 = 2 = 25. For x = 2x + and x = x 2 + x = 3x x = 2x 2 3x 2 x = x + 3 x = 2x+ x 2 Domain:, 2 2, 2. For x = 4x and x = 2x + x = 2x x = 8x 2 + 6x x = 2 6x x = 4x 2x Domain:, 2 2,

7 94 Relations and Functions 3. For x = x 2 and x = 3x + x = x 2 + 3x x = 3x 3 x 2 x = x 2 3x + x = x2 3x Domain:, 3 3, 4. For x = x 2 x and x = 7x + x = x 2 + 6x x = 7x 3 7x 2 x = x 2 8x x = x 7 Domain:, 0 0, 5. For x = x 2 4 and x = 3x x = x 2 + 3x + 2 x = 3x 3 + 6x 2 2x 24 x = x 2 3x 0 x = x 2 3 Domain:, 2 2, 6. For x = x 2 + x + 6 and x = x x = x 3 x = x 4 + x 3 + 5x 2 9x 54 x = 2x 2 + x + 5 x = x+2 x+3 Domain:, 3 3, 3 3, 7. For x = x 2 and x = 2 x + x = x2 +4 2x Domain:, 0 0, x = Domain:, 0 0, x = x2 4 2x Domain:, 0 0, x = x2 4 Domain:, 0 0,

8 .5 Function Arithmetic For x = x and x = x + x = x2 2x+2 x Domain:,, x = Domain:,, x = x2 2x x Domain:,, x = x 2 2x + Domain:,, 9. For x = x and x = x + + x = x + x + Domain: [, x = x x + Domain: [, x = x x + Domain: [, x = x x+ Domain:, 20. For x = x 5 and x = x = x 5 + x = 2 x 5 Domain: [5, x = x 5 Domain: [5, x = 0 Domain: [5, x = Domain: 5, x + 3h 25. 2x h x + 4h 27. 2x h x 2 + 3xh + h m 30. 2ax + ah + b xx + h 32. 2x + h x 2 x + h x 34x + 4h x h x 2 x + 5x + h x + x + h +

9 96 Relations and Functions x 9x + h x + h 9 + x x 4h x a ax + ah + b + ax + b 44. x + h 2/3 + x + h /3 x /3 + x 2/3 2x 2 + 2xh + 2x + h 2x + 2x + 2h + 2 2x + 2h + + 2x + 4 x h + 4 x 3x 2 + 3xh + h 2 x + h 3/2 + x 3/2 46. C0 = 26, so the ixed costs are $26. C0 = 4.6, so when 0 shirts are produced, the cost per shirt is $4.60. p5 = 20, so to sell 5 shirts, set the price at $20 per shirt. Rx = 2x x, 0 x 5 P x = 2x x 26, 0 x 5 P x = 0 when x = and x = 3. These are the break even points, so sellin shirt or 3 shirts will uarantee the revenue earned exactly recoups the cost o production. 47. C0 = 00, so the ixed costs are $00. C0 = 20, so when 0 bottles o tonic are produced, the cost per bottle is $20. p5 = 30, so to sell 5 bottles o tonic, set the price at $30 per bottle. Rx = x x, 0 x 35 P x = x x 00, 0 x 35 P x = 0 when x = 5 and x = 20. These are the break even points, so sellin 5 bottles o tonic or 20 bottles o tonic will uarantee the revenue earned exactly recoups the cost o production. 48. C0 = 240, so the ixed costs are 240 or $2.40. C0 = 42, so when 0 cups o lemonade are made, the cost per cup is 42. p5 = 75, so to sell 5 cups o lemonade, set the price at 75 per cup. Rx = 3x x, 0 x 30 P x = 3x x 240, 0 x 30 P x = 0 when x = 4 and x = 20. These are the break even points, so sellin 4 cups o lemonade or 20 cups o lemonade will uarantee the revenue earned exactly recoups the cost o production.

10 .5 Function Arithmetic C0 = 36, so the daily ixed costs are $36. C0 = 6.6, so when 0 pies are made, the cost per pie is $6.60. p5 = 9.5, so to sell 5 pies a day, set the price at $9.50 per pie. Rx = 0.5x 2 + 2x, 0 x 24 P x = 0.5x 2 + 9x 36, 0 x 24 P x = 0 when x = 6 and x = 2. These are the break even points, so sellin 6 pies or 2 pies a day will uarantee the revenue earned exactly recoups the cost o production. 50. C0 = 000, so the monthly ixed costs are 000 hundred dollars, or $00,000. C0 = 20, so when 0 scooters are made, the cost per scooter is 20 hundred dollars, or $2,000. p5 = 30, so to sell 5 scooters a month, set the price at 30 hundred dollars, or $3,000 per scooter. Rx = 2x x, 0 x 70 P x = 2x x 000, 0 x 70 P x = 0 when x = 0 and x = 50. These are the break even points, so sellin 0 scooters or 50 scooters a month will uarantee the revenue earned exactly recoups the cost o production = = = = = = does not exist 58. = = does not exist 6. 3 = = 2