Function Continuity

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Answer Key & Explanations

Function Continuity · Grade 12 · Worksheet 1

1. Emma is designing a roller coaster track that follows the piecewise function f(x) = { x^2 + 1 for x < 2, ax + b for 2 ≤ x ≤ 4, 3x - 1 for x > 4 }. To ensure a smooth ride, the track must be continuous at both transition points x = 2 and x = 4. What values must the parameters a and b have to guarantee continuity throughout the track? Answer: a=4, b=-3 Solution: For continuity at x = 2, the left-hand limit and right-hand limit must equal f(2).
   Full step-by-step solution

   Step 1: For continuity at x = 2, the left-hand limit and right-hand limit must equal f(2). Step 2: Left-hand limit as x→2⁻: f(x) = x^2 + 1 → 2^2 + 1 = 5 Step 3: Right-hand limit as x→2⁺: f(x) = ax + b → a(2) + b = 2a + b Step 4: For continuity at x = 2: 2a + b = 5 Step 5: For continuity at x = 4, the left-hand limit and right-hand limit must equal f(4). Step 6: Left-hand limit as x→4⁻: f(x) = ax + b → a(4) + b = 4a + b Step 7: Right-hand limit as x→4⁺: f(x) = 3x - 1 → 3(4) - 1 = 11 Step 8: For continuity at x = 4: 4a + b = 11 Step 9: Solve the system of equations: 2a + b = 5 and 4a + b = 11 Step 10: Subtract first equation from second: (4a + b) - (2a + b) = 11 - 5 → 2a = 6 → a = 3 Step 11: Substitute a = 3 into 2a + b = 5: 2(3) + b = 5 → 6 + b = 5 → b = -1 Step 12: The values are a = 3 and b = -1 The answer is a=3, b=-1.

2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (5t^2 * e^(-0.3t))/(t^2 + 1), where t is measured in hours. The researchers need to determine if this concentration function is continuous for all t ≥ 0, particularly at t = 0 where the function appears to have an indeterminate form. Analyze the continuity of C(t) at t = 0 and explain your reasoning mathematically. Answer: The function is continuous at t = 0. Solution: Write the function clearly. C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1) for t ≥ 0. Check if C(0) is defined by direct substitution.
   Full step-by-step solution

   Let's analyze the continuity of C(t) at t = 0 step by step. Step 1: Write the function clearly. C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1) for t ≥ 0. Step 2: Check if C(0) is defined by direct substitution. At t = 0: Numerator: 5 * (0)^2 * e^(-0.3*0) = 5 * 0 * 1 = 0. Denominator: (0)^2 + 1 = 1. So C(0) = 0/1 = 0. The function is defined at t = 0: C(0) = 0. Step 3: Check the limit as t approaches 0. We need to find limit as t -> 0+ of C(t). C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1). As t -> 0: - e^(-0.3 t) -> e^0 = 1. - So numerator ~ 5 t^2 * 1 = 5 t^2. - Denominator = t^2 + 1 -> 1. Thus C(t) ~ (5 t^2) / 1 = 5 t^2 as t -> 0. So limit as t -> 0+ of C(t) = 0. Step 4: Compare the limit and the function value. We have: limit as t -> 0+ of C(t) = 0. C(0) = 0. They are equal. Step 5: Conclusion about continuity. Since: 1. C(0) is defined, 2. The limit as t -> 0+ exists, 3. The limit equals the function value at t = 0, the function C(t) is continuous at t = 0. Final answer: The function is continuous at t = 0.

3. Consider the function f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. Determine the value that f(2) should be assigned to make the function continuous at x = 2. Answer: 4 Solution: f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. We want to assign f(2) so that f is continuous at x = 2. For continuity at x = 2, we need: lim (x → 2) f(x) = f(2).
   Full step-by-step solution

   Let's go step-by-step. We have the function: f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. --- **Step 1: Understand the problem** We want to assign f(2) so that f is continuous at x = 2. For continuity at x = 2, we need: lim (x → 2) f(x) = f(2). So we must find the limit of f(x) as x approaches 2. --- **Step 2: Simplify f(x) for x ≠ 2** Factor the numerator: x^2 - 4 = (x - 2)(x + 2). So for x ≠ 2: f(x) = [(x - 2)(x + 2)] / (x - 2) = x + 2. --- **Step 3: Take the limit** Since f(x) = x + 2 for all x ≠ 2, lim (x → 2) f(x) = lim (x → 2) (x + 2) = 2 + 2 = 4. --- **Step 4: Assign f(2)** For continuity, f(2) must equal the limit: f(2) = 4. --- **Final answer:** 4

4. Liam is analyzing the continuity of a function f(x) defined piecewise as f(x) = { x² - 4 for x < 2, ax + b for 2 ≤ x ≤ 3, 2x - 1 for x > 3 }. He wants the function to be continuous at both x = 2 and x = 3. What values must the parameters a and b have to ensure continuity at both points? Answer: a=3, b=-2 Solution: Continuity at a point requires that the limit from the left equals the limit from the right, and both equal the function value at that point.
   Full step-by-step solution

   Continuity at a point requires that the limit from the left equals the limit from the right, and both equal the function value at that point. For piecewise functions, this often creates a system of equations that can be solved for unknown parameters. The concept applies to any function with multiple definitions across different intervals.

5. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (5t^2 * e^(-0.3t))/(t^3 + 1) for t ≥ 0, where t is measured in hours. The researchers need to determine if this function is continuous at t = 2 hours, which represents a critical metabolic transition point. Analyze the continuity of C(t) at t = 2. Answer: The function is continuous at t = 2 Solution: Recall the definition of continuity at a point. A function f(t) is continuous at t = a if: 1. f(a) is defined 2.
   Full step-by-step solution

   Let's analyze the continuity of C(t) at t = 2. Step 1: Recall the definition of continuity at a point. A function f(t) is continuous at t = a if: 1. f(a) is defined 2. The limit of f(t) as t approaches a exists 3. The limit equals f(a) So for C(t) to be continuous at t = 2, we need: lim(t→2) C(t) = C(2) Step 2: Check if C(2) is defined. C(t) = (5t^2 * e^(-0.3t))/(t^3 + 1) Substitute t = 2: C(2) = (5*(2)^2 * e^(-0.3*2))/((2)^3 + 1) = (5*4 * e^(-0.6))/(8 + 1) = (20 * e^(-0.6))/9 Since e^(-0.6) is a defined positive number, and we're dividing by 9 (not zero), C(2) is defined. Step 3: Find the limit as t approaches 2. We need to find lim(t→2) C(t) = lim(t→2) (5t^2 * e^(-0.3t))/(t^3 + 1) Since this is a rational function where the denominator is not zero at t = 2, we can use direct substitution: lim(t→2) (5t^2 * e^(-0.3t))/(t^3 + 1) = (5*(2)^2 * e^(-0.3*2))/((2)^3 + 1) = (5*4 * e^(-0.6))/(8 + 1) = (20 * e^(-0.6))/9 Step 4: Compare the limit and the function value. From Step 2: C(2) = (20 * e^(-0.6))/9 From Step 3: lim(t→2) C(t) = (20 * e^(-0.6))/9 Since both values are equal, all three conditions for continuity are satisfied. Step 5: Conclusion. The function C(t) is continuous at t = 2 because: - C(2) is defined - The limit as t approaches 2 exists - The limit equals C(2) Therefore, the function is continuous at the critical metabolic transition point t = 2 hours.

6. An environmental engineer is modeling the temperature distribution in a lake using the piecewise function T(x) = { 2x + 5 for x < 3, ax² + bx for 3 ≤ x ≤ 5, 4x - 7 for x > 5 }, where x represents distance from the shore in kilometers and T(x) is temperature in degrees Celsius. To ensure the model accurately represents the gradual temperature change throughout the lake, the function must be continuous at both transition points x = 3 km and x = 5 km. What values must the parameters a and b have to ensure continuity throughout the lake? Answer: a=1, b=-1 Solution: For continuity at x = 3, the left-hand limit and right-hand limit must be equal.
   Full step-by-step solution

   Step 1: For continuity at x = 3, the left-hand limit and right-hand limit must be equal. Left-hand limit (x→3⁻): T(x) = 2x + 5 → 2(3) + 5 = 6 + 5 = 11 Right-hand limit (x→3⁺): T(x) = ax² + bx → a(3)² + b(3) = 9a + 3b Set equal: 9a + 3b = 11 Step 2: For continuity at x = 5, the left-hand limit and right-hand limit must be equal. Left-hand limit (x→5⁻): T(x) = ax² + bx → a(5)² + b(5) = 25a + 5b Right-hand limit (x→5⁺): T(x) = 4x - 7 → 4(5) - 7 = 20 - 7 = 13 Set equal: 25a + 5b = 13 Step 3: Solve the system of equations: Equation 1: 9a + 3b = 11 Equation 2: 25a + 5b = 13 Step 4: Multiply Equation 1 by 5: 45a + 15b = 55 Multiply Equation 2 by 3: 75a + 15b = 39 Step 5: Subtract the modified equations: (75a + 15b) - (45a + 15b) = 39 - 55 30a = -16 a = -16/30 = -8/15 Step 6: Substitute a = -8/15 into Equation 1: 9(-8/15) + 3b = 11 -72/15 + 3b = 11 -24/5 + 3b = 11 3b = 11 + 24/5 = 55/5 + 24/5 = 79/5 b = 79/15 Step 7: Verify with Equation 2: 25(-8/15) + 5(79/15) = -200/15 + 395/15 = 195/15 = 13 ✓ The answer is a = -8/15, b = 79/15.