Function Operations

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Function Operations · Grade 12 · Worksheet 3

1. Emma is a water quality analyst studying two pollutants in a river. The concentration of pollutant A (in parts per million) over time t (in years) is modeled by the function A(t) = 5t^2 + 20t, and the concentration of pollutant B is modeled by B(t) = 15t + 10. Emma needs to determine the function that represents the total concentration of both pollutants combined, T(t) = A(t) + B(t). She then plans to use the quotient function R(t) = T(t) / B(t) to analyze the ratio of total pollutants to pollutant B. Find both T(t) and R(t), and simplify R(t) as much as possible. Answer: T(t) = 5t^2 + 35t + 10, R(t) = (5t^2 + 35t + 10) / (15t + 10) Solution: Find T(t) = A(t) + B(t). A(t) = 5t^2 + 20t B(t) = 15t + 10 T(t) = (5t^2 + 20t) + (15t + 10) = 5t^2 + (20t + 15t) + 10 = 5t^2 + 35t + 10 Find R(t) = T(t) / B(t).
   Full step-by-step solution

   Step 1: Find T(t) = A(t) + B(t). A(t) = 5t^2 + 20t B(t) = 15t + 10 T(t) = (5t^2 + 20t) + (15t + 10) = 5t^2 + (20t + 15t) + 10 = 5t^2 + 35t + 10 Step 2: Find R(t) = T(t) / B(t). R(t) = (5t^2 + 35t + 10) / (15t + 10) Step 3: Check for simplification. Factor the numerator: 5t^2 + 35t + 10 = 5(t^2 + 7t + 2). Factor the denominator: 15t + 10 = 5(3t + 2). So R(t) = 5(t^2 + 7t + 2) / [5(3t + 2)] = (t^2 + 7t + 2) / (3t + 2). The quadratic t^2 + 7t + 2 does not factor with integer roots, so no further simplification is possible. The answer is T(t) = 5t^2 + 35t + 10 and R(t) = (t^2 + 7t + 2) / (3t + 2).

2. f(x) = 7x³ - 12x² + 17, g(x) = 2x³ + 7x - 22. Find (f + g)(x) = ? Answer: 9x³ - 12x² + 7x - 5 Solution: Write the sum: (f + g)(x) = (7x³ - 12x² + 17) + (2x³ + 7x - 22) Group like terms: (7x³ + 2x³) + (-12x²) + (7x) + (17 - 22) Add the x³ terms: 7 + 2 = 9, so 9x³ The -12x² term has no like term, so it remains -12x² The 7x term has no like term, so it remains 7x Add the constants: 17 + (-22) = -5…
   Full step-by-step solution

   Step 1: Write the sum: (f + g)(x) = (7x³ - 12x² + 17) + (2x³ + 7x - 22) Step 2: Group like terms: (7x³ + 2x³) + (-12x²) + (7x) + (17 - 22) Step 3: Add the x³ terms: 7 + 2 = 9, so 9x³ Step 4: The -12x² term has no like term, so it remains -12x² Step 5: The 7x term has no like term, so it remains 7x Step 6: Add the constants: 17 + (-22) = -5 Step 7: Combine all terms: 9x³ - 12x² + 7x - 5 The answer is 9x³ - 12x² + 7x - 5.

3. f(x) = 2x² - 7x + 3, g(x) = x² + 2x - 5. Find (f + g)(x) = ? Answer: 3x² - 5x - 2 Solution: Write out the sum: (f + g)(x) = (2x² - 7x + 3) + (x² + 2x - 5) Combine like terms for x²: 2x² + x² = 3x² Combine like terms for x: -7x + 2x = -5x Combine constant terms: 3 + (-5) = -2 Write the final expression: 3x² - 5x - 2 The answer is 3x² - 5x - 2.
   Full step-by-step solution

   Step 1: Write out the sum: (f + g)(x) = (2x² - 7x + 3) + (x² + 2x - 5) Step 2: Combine like terms for x²: 2x² + x² = 3x² Step 3: Combine like terms for x: -7x + 2x = -5x Step 4: Combine constant terms: 3 + (-5) = -2 Step 5: Write the final expression: 3x² - 5x - 2 The answer is 3x² - 5x - 2.

4. Emma is a city planner analyzing two functions that model traffic flow in a downtown district. The number of vehicles entering the district per hour is modeled by E(x) = 2x^2 + 15x, where x is the hour of the day (from 0 to 12). The number of vehicles leaving the district per hour is modeled by L(x) = x^2 + 5x + 50. Emma needs to find the function N(x) = E(x) - L(x), which represents the net number of vehicles remaining in the district per hour. What is N(x)? Answer: x^2 + 10x - 50 Solution: Write the subtraction: N(x) = E(x) - L(x) = (2x^2 + 15x) - (x^2 + 5x + 50) Distribute the negative sign to each term in L(x): N(x) = 2x^2 + 15x - x^2 - 5x - 50 Combine like terms: 2x^2 - x^2 = x^2; 15x - 5x = 10x; constant term is -50 N(x) = x^2 + 10x - 50 The answer is N(x) = x^2 + 10x - 50.
   Full step-by-step solution

   Step 1: Write the subtraction: N(x) = E(x) - L(x) = (2x^2 + 15x) - (x^2 + 5x + 50) Step 2: Distribute the negative sign to each term in L(x): N(x) = 2x^2 + 15x - x^2 - 5x - 50 Step 3: Combine like terms: 2x^2 - x^2 = x^2; 15x - 5x = 10x; constant term is -50 Step 4: N(x) = x^2 + 10x - 50 The answer is N(x) = x^2 + 10x - 50.

5. A marine biologist is studying the population growth of an endangered coral species. The population function is P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)), where t is time in years since monitoring began. The conservation effectiveness is modeled by E(p) = ln(p/500), where p is the population size. What is the rate of change of conservation effectiveness with respect to time when t = 20 years? Answer: 0.015 Solution: We need to find dE/dt when t = 20. Since E is a function of p and p is a function of t, we use the chain rule: dE/dt = (dE/dp) × (dp/dt). First, find dE/dp.
   Full step-by-step solution

   Step 1: We need to find dE/dt when t = 20. Since E is a function of p and p is a function of t, we use the chain rule: dE/dt = (dE/dp) × (dp/dt). Step 2: First, find dE/dp. E(p) = ln(p/500) = ln(p) - ln(500). So dE/dp = 1/p. Step 3: Now find dp/dt. P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)). Using the quotient rule: dp/dt = [2000(0.03)e^(0.03t)(1 + 0.5e^(0.03t)) - 2000e^(0.03t)(0.5)(0.03)e^(0.03t)] / (1 + 0.5e^(0.03t))^2. Step 4: Simplify: dp/dt = [60e^(0.03t)(1 + 0.5e^(0.03t)) - 30e^(0.06t)] / (1 + 0.5e^(0.03t))^2. Step 5: When t = 20, first calculate e^(0.03×20) = e^(0.6) ≈ 1.8221. Then p(20) = 2000×1.8221 / (1 + 0.5×1.8221) = 3644.2 / (1 + 0.91105) = 3644.2 / 1.91105 ≈ 1907.5. Step 6: Now calculate dp/dt at t = 20: dp/dt = [60×1.8221×(1 + 0.91105) - 30×(1.8221)^2] / (1.91105)^2 = [109.326×(1.91105) - 30×3.320] / 3.6521 = [208.9 - 99.6] / 3.6521 = 109.3 / 3.6521 ≈ 29.93. Step 7: Now dE/dt = (dE/dp) × (dp/dt) = (1/p) × (dp/dt) = (1/1907.5) × 29.93 ≈ 0.000524 × 29.93 ≈ 0.0157. Step 8: Rounded to three decimal places, the rate of change is 0.015. The answer is 0.015.

6. f(x) = 2x³ - 7x² + 4x - 12, g(x) = 3x² - 2x + 7. Find (f - g)(x) Answer: 2x³ - 10x² + 6x - 19 Solution: Write the expression for (f - g)(x): (2x³ - 7x² + 4x - 12) - (3x² - 2x + 7) Distribute the negative sign: 2x³ - 7x² + 4x - 12 - 3x² + 2x - 7 Combine like terms for x³: 2x³ Combine like terms for x²: -7x² - 3x² = -10x² Combine like terms for x: 4x + 2x = 6x Combine constant terms: -12 - 7 = -19…
   Full step-by-step solution

   Step 1: Write the expression for (f - g)(x): (2x³ - 7x² + 4x - 12) - (3x² - 2x + 7) Step 2: Distribute the negative sign: 2x³ - 7x² + 4x - 12 - 3x² + 2x - 7 Step 3: Combine like terms for x³: 2x³ Step 4: Combine like terms for x²: -7x² - 3x² = -10x² Step 5: Combine like terms for x: 4x + 2x = 6x Step 6: Combine constant terms: -12 - 7 = -19 Step 7: Write the final result: 2x³ - 10x² + 6x - 19

7. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is C(t) = 50te^(-0.2t) mg/L, where t is hours after administration. The therapeutic window for this drug is between 15 mg/L and 35 mg/L. Determine the time interval during which the drug concentration remains within the therapeutic window. Answer: Between approximately 0.36 hours and 5.83 hours Solution: When analyzing when a function remains within a specific range, we find the intersection points where the function equals the boundary values.
   Full step-by-step solution

   When analyzing when a function remains within a specific range, we find the intersection points where the function equals the boundary values. For drug concentration models, this helps determine effective treatment duration. The process involves setting up equations and using algebraic techniques to solve for the time variable, which often requires applying inverse operations to exponential functions.