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Function Operations

Grade 12 · Algebra · Worksheet 3

  1. Tane is a wildlife biologist studying the population of two competing bird species on an island. The population of the Kākā (in hundreds) is modeled by the function K(t) = 3t² + 5, and the population of the Tūī (in hundreds) is modeled by T(t) = 7t - 9, where t is the number of years since the study began. Tane needs to determine the combined population function, C(t) = (K + T)(t), to analyze the total bird population. He also needs to find the difference function, D(t) = (K - T)(t), to understand the population advantage of one species over the other. Determine both C(t) and D(t), and calculate the combined population (in hundreds) when t = 5 years. Answer: ______________
  2. f(x) = 7x³ - 13x, g(x) = 5x² + 9. Find (f + g)(x) Answer: ______________
  3. 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: ______________
  4. f(x) = 3x² - 5x + 10, g(x) = 2x² + 5x - 15. Find (f - g)(x) = ? Answer: ______________
  5. f(x) = 3x³ - 5x + 7, g(x) = 2x³ + 9x - 11. Find (f+g)(x) = ? Answer: ______________
  6. f(x) = 6x³ - 11x + 4, g(x) = 2x² + 5x - 7. Find (f - g)(x) = ? Answer: ______________
  7. An environmental scientist is modeling the population growth of an endangered species using the function P(t) = 2000 / (1 + 9e^(-0.3t)), where t is time in years since conservation efforts began. The carrying capacity of the habitat is modeled by C(P) = 5000 - 0.2P. Determine the composite function C(P(t)) and calculate the habitat's carrying capacity when the population reaches 1500 individuals. Answer: ______________
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Answer Key & Explanations

Function Operations · Grade 12 · Worksheet 3

  1. Tane is a wildlife biologist studying the population of two competing bird species on an island. The population of the Kākā (in hundreds) is modeled by the function K(t) = 3t² + 5, and the population of the Tūī (in hundreds) is modeled by T(t) = 7t - 9, where t is the number of years since the study began. Tane needs to determine the combined population function, C(t) = (K + T)(t), to analyze the total bird population. He also needs to find the difference function, D(t) = (K - T)(t), to understand the population advantage of one species over the other. Determine both C(t) and D(t), and calculate the combined population (in hundreds) when t = 5 years. Answer: C(t) = 3t² + 7t - 4, D(t) = 3t² - 7t + 14, C(5) = 106 Solution: Write the given functions. K(t) = 3t² + 5, T(t) = 7t - 9. Find the combined population function C(t) = (K + T)(t).
    Full step-by-step solution

    Step 1: Write the given functions. K(t) = 3t² + 5, T(t) = 7t - 9. Step 2: Find the combined population function C(t) = (K + T)(t). C(t) = K(t) + T(t) C(t) = (3t² + 5) + (7t - 9) C(t) = 3t² + 7t + 5 - 9 C(t) = 3t² + 7t - 4 Step 3: Find the difference function D(t) = (K - T)(t). D(t) = K(t) - T(t) D(t) = (3t² + 5) - (7t - 9) D(t) = 3t² + 5 - 7t + 9 D(t) = 3t² - 7t + 14 Step 4: Calculate the combined population when t = 5 years. C(5) = 3(5)² + 7(5) - 4 C(5) = 3(25) + 35 - 4 C(5) = 75 + 35 - 4 C(5) = 106 Step 5: Interpret the result. C(5) = 106 means the combined population of Kākā and Tūī after 5 years is 106 hundred birds, or 10,600 birds. The answer is C(t) = 3t² + 7t - 4, D(t) = 3t² - 7t + 14, C(5) = 106.

  2. f(x) = 7x³ - 13x, g(x) = 5x² + 9. Find (f + g)(x) Answer: 7x³ + 5x² - 13x + 9 Solution: Write the sum of the functions: (f + g)(x) = (7x³ - 13x) + (5x² + 9) Remove parentheses: 7x³ - 13x + 5x² + 9 Rearrange in standard form (descending order): 7x³ + 5x² - 13x + 9 The final answer is 7x³ + 5x² - 13x + 9.
    Full step-by-step solution

    Step 1: Write the sum of the functions: (f + g)(x) = (7x³ - 13x) + (5x² + 9) Step 2: Remove parentheses: 7x³ - 13x + 5x² + 9 Step 3: Rearrange in standard form (descending order): 7x³ + 5x² - 13x + 9 Step 4: Verify there are no like terms to combine further The final answer is 7x³ + 5x² - 13x + 9.

  3. 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.

  4. f(x) = 3x² - 5x + 10, g(x) = 2x² + 5x - 15. Find (f - g)(x) = ? Answer: x² - 10x + 25 Solution: Write the expression for (f - g)(x) = f(x) - g(x) Substitute the functions: (3x² - 5x + 10) - (2x² + 5x - 15) Distribute the negative sign: 3x² - 5x + 10 - 2x² - 5x + 15 - x² terms: 3x² - 2x² = x² - x terms: -5x - 5x = -10x - constant terms: 10 + 15 = 25 Write the final expression: x² - 10x + 25…
    Full step-by-step solution

    Step 1: Write the expression for (f - g)(x) = f(x) - g(x) Step 2: Substitute the functions: (3x² - 5x + 10) - (2x² + 5x - 15) Step 3: Distribute the negative sign: 3x² - 5x + 10 - 2x² - 5x + 15 Step 4: Combine like terms: - x² terms: 3x² - 2x² = x² - x terms: -5x - 5x = -10x - constant terms: 10 + 15 = 25 Step 5: Write the final expression: x² - 10x + 25 The answer is x² - 10x + 25.

  5. f(x) = 3x³ - 5x + 7, g(x) = 2x³ + 9x - 11. Find (f+g)(x) = ? Answer: 5x³ + 4x - 4 Solution: Write the sum of the functions: (f+g)(x) = f(x) + g(x) = (3x³ - 5x + 7) + (2x³ + 9x - 11) Combine x³ terms: 3x³ + 2x³ = 5x³ Combine x terms: -5x + 9x = 4x Combine constant terms: 7 + (-11) = -4 Write the final result: 5x³ + 4x - 4 The answer is 5x³ + 4x - 4.
    Full step-by-step solution

    Step 1: Write the sum of the functions: (f+g)(x) = f(x) + g(x) = (3x³ - 5x + 7) + (2x³ + 9x - 11) Step 2: Combine x³ terms: 3x³ + 2x³ = 5x³ Step 3: Combine x terms: -5x + 9x = 4x Step 4: Combine constant terms: 7 + (-11) = -4 Step 5: Write the final result: 5x³ + 4x - 4 The answer is 5x³ + 4x - 4.

  6. f(x) = 6x³ - 11x + 4, g(x) = 2x² + 5x - 7. Find (f - g)(x) = ? Answer: 6x³ - 2x² - 16x + 11 Solution: Write out (f - g)(x) = f(x) - g(x) Substitute the functions: (6x³ - 11x + 4) - (2x² + 5x - 7) Distribute the negative sign: 6x³ - 11x + 4 - 2x² - 5x + 7 - x³ terms: 6x³ - x² terms: -2x² - x terms: -11x - 5x = -16x - constant terms: 4 + 7 = 11 Write the final expression: 6x³ - 2x² - 16x + 11
    Full step-by-step solution

    Step 1: Write out (f - g)(x) = f(x) - g(x) Step 2: Substitute the functions: (6x³ - 11x + 4) - (2x² + 5x - 7) Step 3: Distribute the negative sign: 6x³ - 11x + 4 - 2x² - 5x + 7 Step 4: Combine like terms: - x³ terms: 6x³ - x² terms: -2x² - x terms: -11x - 5x = -16x - constant terms: 4 + 7 = 11 Step 5: Write the final expression: 6x³ - 2x² - 16x + 11

  7. An environmental scientist is modeling the population growth of an endangered species using the function P(t) = 2000 / (1 + 9e^(-0.3t)), where t is time in years since conservation efforts began. The carrying capacity of the habitat is modeled by C(P) = 5000 - 0.2P. Determine the composite function C(P(t)) and calculate the habitat's carrying capacity when the population reaches 1500 individuals. Answer: 4700 Solution: P(t) = 2000 / (1 + 9e^(-0.3t)) C(P) = 5000 - 0.2P C(P(t)) = 5000 - 0.2 × [2000 / (1 + 9e^(-0.3t))] C(P(t)) = 5000 - 400 / (1 + 9e^(-0.3t)) We need to find when P(t) = 1500 1500 = 2000 / (1 + 9e^(-0.3t)) 1 + 9e^(-0.3t) = 2000 / 1500 1 + 9e^(-0.3t) = 4/3 9e^(-0.3t) = 4/3 - 1 9e^(-0.3t) = 1/3…
    Full step-by-step solution

    Step 1: Write the given functions P(t) = 2000 / (1 + 9e^(-0.3t)) C(P) = 5000 - 0.2P Step 2: Create the composite function C(P(t)) C(P(t)) = 5000 - 0.2 × [2000 / (1 + 9e^(-0.3t))] C(P(t)) = 5000 - 400 / (1 + 9e^(-0.3t)) Step 3: We need to find when P(t) = 1500 1500 = 2000 / (1 + 9e^(-0.3t)) Step 4: Solve for the expression with t 1 + 9e^(-0.3t) = 2000 / 1500 1 + 9e^(-0.3t) = 4/3 9e^(-0.3t) = 4/3 - 1 9e^(-0.3t) = 1/3 e^(-0.3t) = 1/27 Step 5: Substitute back into C(P(t)) C(P(t)) = 5000 - 400 / (1 + 9e^(-0.3t)) We know 1 + 9e^(-0.3t) = 4/3 C(P(t)) = 5000 - 400 / (4/3) C(P(t)) = 5000 - 400 × (3/4) C(P(t)) = 5000 - 300 C(P(t)) = 4700 The habitat's carrying capacity when the population reaches 1500 individuals is 4700.