Function Operations

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Function Operations · Grade 12 · Worksheet 2

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) = 2x² - 7x + 3, g(x) = x² + 2x - 5. Find (f - g)(x) = ? Answer: x² - 9x + 8 Solution: (f - g)(x) = f(x) - g(x) (f - g)(x) = (2x² - 7x + 3) - (x² + 2x - 5) (f - g)(x) = 2x² - 7x + 3 - x² - 2x + 5 For x² terms: 2x² - x² = x² For x terms: -7x - 2x = -9x For constant terms: 3 + 5 = 8 (f - g)(x) = x² - 9x + 8 The answer is x² - 9x + 8.
   Full step-by-step solution

   Step 1: Write out the expression for (f - g)(x) (f - g)(x) = f(x) - g(x) Step 2: Substitute the given functions (f - g)(x) = (2x² - 7x + 3) - (x² + 2x - 5) Step 3: Distribute the negative sign to all terms of g(x) (f - g)(x) = 2x² - 7x + 3 - x² - 2x + 5 Step 4: Combine like terms For x² terms: 2x² - x² = x² For x terms: -7x - 2x = -9x For constant terms: 3 + 5 = 8 Step 5: Write the final expression (f - g)(x) = x² - 9x + 8 The answer is x² - 9x + 8.

3. f(x) = 7x³ - 11x² + 4x - 8, g(x) = 3x³ + 9x - 15. Find (f - g)(x) Answer: 4x³ - 11x² - 5x + 7 Solution: Write the expression for (f - g)(x): (7x³ - 11x² + 4x - 8) - (3x³ + 9x - 15) Distribute the negative sign: 7x³ - 11x² + 4x - 8 - 3x³ - 9x + 15 Combine like terms for x³: 7x³ - 3x³ = 4x³ The -11x² term has no like terms, so it remains: -11x² Combine like terms for x: 4x - 9x = -5x Combine…
   Full step-by-step solution

   Step 1: Write the expression for (f - g)(x): (7x³ - 11x² + 4x - 8) - (3x³ + 9x - 15) Step 2: Distribute the negative sign: 7x³ - 11x² + 4x - 8 - 3x³ - 9x + 15 Step 3: Combine like terms for x³: 7x³ - 3x³ = 4x³ Step 4: The -11x² term has no like terms, so it remains: -11x² Step 5: Combine like terms for x: 4x - 9x = -5x Step 6: Combine constant terms: -8 + 15 = 7 Step 7: Write the final expression: 4x³ - 11x² - 5x + 7

4. A geometric pattern is formed by the function f(x) = x² - 4x + 3 and its transformation g(x) = f(x-2) + 1. The pattern shows two parabolas on a coordinate plane, where g(x) is a horizontal and vertical translation of f(x). What is the vertex of the transformed parabola g(x)? Answer: (3, -2) Solution: Function transformations involve shifting graphs horizontally and vertically.
   Full step-by-step solution

   Function transformations involve shifting graphs horizontally and vertically. For a quadratic function in vertex form, horizontal shifts affect the x-coordinate of the vertex while vertical shifts affect the y-coordinate. Understanding how these transformations combine helps analyze geometric patterns of translated functions.

5. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = 50t * e^(-0.2t), where t is time in hours and C(t) is concentration in mg/L. The therapeutic window requires maintaining concentrations between 40 mg/L and 80 mg/L. Determine the time interval during which the medication remains within this therapeutic range. Answer: Between approximately 1.15 hours and 6.93 hours Solution: When analyzing when a quantity modeled by a function stays within certain bounds, we solve for when the function equals the boundary values.
   Full step-by-step solution

   When analyzing when a quantity modeled by a function stays within certain bounds, we solve for when the function equals the boundary values. For functions involving exponential decay combined with polynomial growth, we often use logarithmic techniques to isolate the variable. This approach is commonly used in pharmacokinetics to determine dosing intervals and ensure medications remain effective without reaching toxic levels.

6. Mere is an environmental engineer analyzing two pollution control models for a factory. The rate of pollutant removal by a chemical scrubber is modeled by the function f(x) = 4x^2 + 8x, where x is the concentration of pollutant in parts per million (ppm). The rate of pollutant removal by a biological filter is modeled by the function g(x) = 2x + 6, where x is the same concentration. To determine the combined removal rate when both systems operate simultaneously, Mere needs to find the product of these two functions. What is the function h(x) = f(x) * g(x) that represents the combined removal rate? Answer: 8x^3 + 40x^2 + 48x Solution: Write the product h(x) = f(x) * g(x) = (4x^2 + 8x) * (2x + 6). Use the distributive property. Multiply each term in the first polynomial by each term in the second polynomial.
   Full step-by-step solution

   Step 1: Write the product h(x) = f(x) * g(x) = (4x^2 + 8x) * (2x + 6). Step 2: Use the distributive property. Multiply each term in the first polynomial by each term in the second polynomial. Step 3: First, multiply 4x^2 by 2x to get 8x^3. Step 4: Then, multiply 4x^2 by 6 to get 24x^2. Step 5: Next, multiply 8x by 2x to get 16x^2. Step 6: Then, multiply 8x by 6 to get 48x. Step 7: Now add all terms: 8x^3 + 24x^2 + 16x^2 + 48x. Step 8: Combine like terms: 24x^2 + 16x^2 = 40x^2. Step 9: The final function is h(x) = 8x^3 + 40x^2 + 48x. The answer is 8x^3 + 40x^2 + 48x.