Piecewise Functions

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Answer Key & Explanations

Piecewise Functions · Grade 12 · Worksheet 1

1. Kaia is a civil engineer designing a water drainage system for a new housing development. The water flow rate (in liters per second) through a drainage pipe is modeled by the piecewise function f(x) = { 3x + 1 for -3 ≤ x < 1; -x² + 5x - 1 for 1 ≤ x ≤ 5 }, where x represents the time in hours after a storm begins. Kaia needs to evaluate the flow rate at specific times to check for overflow risks. Evaluate f(-3), f(1), f(3), and f(5). Then, describe the behavior of the graph at x = 1, specifically whether there is a jump, break, or continuity, and identify any other discontinuities in the domain. Answer: f(-3) = -8, f(1) = 3, f(3) = 5, f(5) = -1; Jump discontinuity at x = 1 (left limit = 4, right value = 3); No other discontinuities Solution: Evaluate f(-3). The point x = -3 is in the first piece's domain (-3 ≤ x < 1). Use f(x) = 3x + 1.
   Full step-by-step solution

   Step 1: Evaluate f(-3). The point x = -3 is in the first piece's domain (-3 ≤ x < 1). Use f(x) = 3x + 1. f(-3) = 3(-3) + 1 = -9 + 1 = -8. Step 2: Evaluate f(1). The point x = 1 is in the second piece's domain (1 ≤ x ≤ 5). Use f(x) = -x² + 5x - 1. f(1) = -(1)² + 5(1) - 1 = -1 + 5 - 1 = 3. Step 3: Evaluate f(3). x = 3 is in the second piece's domain. f(3) = -(3)² + 5(3) - 1 = -9 + 15 - 1 = 5. Step 4: Evaluate f(5). x = 5 is in the second piece's domain. f(5) = -(5)² + 5(5) - 1 = -25 + 25 - 1 = -1. Step 5: Analyze behavior at x = 1. The left-hand limit as x approaches 1 from the first piece: limit = 3(1) + 1 = 4 (since x = 1 is not included in the first piece, this is the limit). The right-hand value at x = 1 from the second piece is f(1) = 3. Since 4 ≠ 3, there is a jump discontinuity at x = 1. Step 6: Check for other discontinuities. The first piece is defined on [-3, 1) and the second on [1, 5]. There are no breaks within each piece as they are polynomial functions, so the only discontinuity is at x = 1. Final answer: f(-3) = -8, f(1) = 3, f(3) = 5, f(5) = -1. There is a jump discontinuity at x = 1 (left limit 4, right value 3). No other discontinuities.

2. Sophia is analyzing a step function defined as g(x) = { 12 for -7 ≤ x < -4; 8 for -4 ≤ x < 1; -3 for 1 ≤ x ≤ 7 }. On a coordinate plane, graph this function and evaluate g(-5), g(-4), g(0), and g(7). Answer: g(-5)=12, g(-4)=8, g(0)=8, g(7)=-3 Solution: Identify the three intervals and their corresponding constant values: - Interval 1: -7 ≤ x < -4 gives g(x) = 12 - Interval 2: -4 ≤ x < 1 gives g(x) = 8 - Interval 3: 1 ≤ x ≤ 7 gives g(x) = -3 Evaluate g(-5): Since -7 ≤ -5 < -4, g(-5) = 12.
   Full step-by-step solution

   Step 1: Identify the three intervals and their corresponding constant values: - Interval 1: -7 ≤ x < -4 gives g(x) = 12 - Interval 2: -4 ≤ x < 1 gives g(x) = 8 - Interval 3: 1 ≤ x ≤ 7 gives g(x) = -3 Step 2: Evaluate g(-5): Since -7 ≤ -5 < -4, g(-5) = 12. Step 3: Evaluate g(-4): Since -4 ≤ -4 < 1 (the inequality is ≤ on the left), g(-4) = 8. Step 4: Evaluate g(0): Since -4 ≤ 0 < 1, g(0) = 8. Step 5: Evaluate g(7): Since 1 ≤ 7 ≤ 7, g(7) = -3. Step 6: Graphically, the function consists of three horizontal line segments: a segment at y = 12 from x = -7 to just before x = -4 (with a closed circle at x = -7 and an open circle at x = -4), a segment at y = 8 from x = -4 to just before x = 1 (with a closed circle at x = -4 and an open circle at x = 1), and a segment at y = -3 from x = 1 to x = 7 (with closed circles at both endpoints). The answer is: g(-5) = 12, g(-4) = 8, g(0) = 8, g(7) = -3.

3. f(x) = {x^2 - 11 if x < -4; 3x + 10 if -4 ≤ x < 2; floor(x) + 7 if x ≥ 2}. Evaluate f(-5), f(-4), f(1), f(2), f(6) Answer: f(-5) = 14, f(-4) = -2, f(1) = 13, f(2) = 9, f(6) = 13 Solution: Evaluate f(-5). Since -5 < -4, use the first piece: f(x) = x^2 - 11. f(-5) = (-5)^2 - 11 = 25 - 11 = 14.
   Full step-by-step solution

   Step 1: Evaluate f(-5). Since -5 < -4, use the first piece: f(x) = x^2 - 11. f(-5) = (-5)^2 - 11 = 25 - 11 = 14. Step 2: Evaluate f(-4). Since -4 ≤ -4 < 2, use the second piece: f(x) = 3x + 10. f(-4) = 3(-4) + 10 = -12 + 10 = -2. Step 3: Evaluate f(1). Since -4 ≤ 1 < 2, use the second piece: f(x) = 3x + 10. f(1) = 3(1) + 10 = 3 + 10 = 13. Step 4: Evaluate f(2). Since 2 ≥ 2, use the third piece: f(x) = floor(x) + 7. floor(2) = 2, so f(2) = 2 + 7 = 9. Step 5: Evaluate f(6). Since 6 ≥ 2, use the third piece: f(x) = floor(x) + 7. floor(6) = 6, so f(6) = 6 + 7 = 13. Final answers: f(-5) = 14, f(-4) = -2, f(1) = 13, f(2) = 9, f(6) = 13.

4. f(x) = {3x + 5 if x < -1; 7 if -1 ≤ x < 3; x² - 2x + 1 if x ≥ 3}. Evaluate f(-2), f(0), f(3), f(4) Answer: f(-2) = -1, f(0) = 7, f(3) = 4, f(4) = 9 Solution: Evaluate f(-2) Since -2 < -1, use the first piece: f(x) = 3x + 5 f(-2) = 3(-2) + 5 = -6 + 5 = -1 Evaluate f(0) Since -1 ≤ 0 < 3, use the second piece: f(x) = 7 f(0) = 7 Evaluate f(3) Since 3 ≥ 3, use the third piece: f(x) = x² - 2x + 1 f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4 Evaluate f(4) Since 4…
   Full step-by-step solution

   Step 1: Evaluate f(-2) Since -2 < -1, use the first piece: f(x) = 3x + 5 f(-2) = 3(-2) + 5 = -6 + 5 = -1 Step 2: Evaluate f(0) Since -1 ≤ 0 < 3, use the second piece: f(x) = 7 f(0) = 7 Step 3: Evaluate f(3) Since 3 ≥ 3, use the third piece: f(x) = x² - 2x + 1 f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4 Step 4: Evaluate f(4) Since 4 ≥ 3, use the third piece: f(x) = x² - 2x + 1 f(4) = (4)² - 2(4) + 1 = 16 - 8 + 1 = 9 Final answer: f(-2) = -1, f(0) = 7, f(3) = 4, f(4) = 9

5. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration function is piecewise defined as C(t) = { 2t^2 - 8t + 10 for 0 ≤ t ≤ 3; -t^2 + 10t - 11 for 3 < t ≤ 8 } where t is hours after administration and C(t) is in milligrams per liter. Determine the time intervals during which the medication concentration is increasing and the intervals during which it is decreasing. Answer: Increasing: (0,2) and (3,5); Decreasing: (2,3) and (5,8) Solution: For continuous functions, we can determine intervals of increase and decrease by analyzing the derivative. When the derivative is positive, the function is increasing; when negative, it's decreasing.
   Full step-by-step solution

   For continuous functions, we can determine intervals of increase and decrease by analyzing the derivative. When the derivative is positive, the function is increasing; when negative, it's decreasing. With piecewise functions, we must check each piece independently and consider the behavior at the boundaries between pieces. This approach is useful in many real-world applications like tracking drug concentrations, population growth, or temperature changes over time.

6. f(x) = {5x + 3 if x < -1; -x^2 + 9 if -1 ≤ x < 5; 11 if x ≥ 5}. Evaluate f(-3), f(-1), f(3), f(5), f(7). Answer: f(-3) = -12, f(-1) = 8, f(3) = 0, f(5) = 11, f(7) = 11 Solution: Evaluate f(-3). Since -3 < -1, use the first piece: f(x) = 5x + 3. f(-3) = 5(-3) + 3 = -15 + 3 = -12.
   Full step-by-step solution

   Step 1: Evaluate f(-3). Since -3 < -1, use the first piece: f(x) = 5x + 3. f(-3) = 5(-3) + 3 = -15 + 3 = -12. Step 2: Evaluate f(-1). Since -1 ≥ -1 and -1 < 5, use the second piece: f(x) = -x^2 + 9. f(-1) = -(-1)^2 + 9 = -1 + 9 = 8. Step 3: Evaluate f(3). Since -1 ≤ 3 < 5, use the second piece: f(x) = -x^2 + 9. f(3) = -(3)^2 + 9 = -9 + 9 = 0. Step 4: Evaluate f(5). Since 5 ≥ 5, use the third piece: f(x) = 11. f(5) = 11. Step 5: Evaluate f(7). Since 7 ≥ 5, use the third piece: f(x) = 11. f(7) = 11. The final answers are: f(-3) = -12, f(-1) = 8, f(3) = 0, f(5) = 11, f(7) = 11.

7. f(x) = {3x - 11 if x < 4; 2x^2 - 5 if 4 ≤ x < 9; 13 if x ≥ 9}. Evaluate f(3), f(4), f(8), f(9), f(12) Answer: f(3) = -2, f(4) = 27, f(8) = 123, f(9) = 13, f(12) = 13 Solution: Evaluate f(3). Since 3 < 4, use the first piece: f(x) = 3x - 11. f(3) = 3(3) - 11 = 9 - 11 = -2.
   Full step-by-step solution

   Step 1: Evaluate f(3). Since 3 < 4, use the first piece: f(x) = 3x - 11. f(3) = 3(3) - 11 = 9 - 11 = -2. Step 2: Evaluate f(4). Since 4 ≤ 4 < 9, use the second piece: f(x) = 2x^2 - 5. f(4) = 2(4)^2 - 5 = 2(16) - 5 = 32 - 5 = 27. Step 3: Evaluate f(8). Since 4 ≤ 8 < 9, use the second piece: f(x) = 2x^2 - 5. f(8) = 2(8)^2 - 5 = 2(64) - 5 = 128 - 5 = 123. Step 4: Evaluate f(9). Since 9 ≥ 9, use the third piece: f(x) = 13. f(9) = 13. Step 5: Evaluate f(12). Since 12 ≥ 9, use the third piece: f(x) = 13. f(12) = 13. Final answer: f(3) = -2, f(4) = 27, f(8) = 123, f(9) = 13, f(12) = 13.