Function Operations

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Function Operations · Grade 12 · Worksheet 2

1. Kaia is a wildlife ecologist studying two species of native birds in a forest reserve. The population of the kākā (in hundreds) after t years is modeled by f(t) = 3t² + 7t + 1, and the population of the kea (in hundreds) is modeled by g(t) = 5t + 9. To understand the combined population dynamics, Kaia defines the total population function T(t) = (f + g)(t). She also needs to find the ratio function R(t) = (f ÷ g)(t) to analyze how the kākā population compares to the kea population over time. Determine T(t) in simplified form, and find R(t) in simplified form, stating any restrictions on the domain for R(t) that make sense in this ecological context. Answer: T(t) = 3t² + 12t + 10; R(t) = (3t² + 7t + 1)/(5t + 9), domain: t >= 0, t ≠ -9/5 Solution: Write the given functions: f(t) = 3t² + 7t + 1, g(t) = 5t + 9. To find T(t) = (f + g)(t), add the expressions: T(t) = (3t² + 7t + 1) + (5t + 9). Combine like terms: The t² term: 3t².
   Full step-by-step solution

   Step 1: Write the given functions: f(t) = 3t² + 7t + 1, g(t) = 5t + 9. Step 2: To find T(t) = (f + g)(t), add the expressions: T(t) = (3t² + 7t + 1) + (5t + 9). Step 3: Combine like terms: The t² term: 3t². The t terms: 7t + 5t = 12t. The constant terms: 1 + 9 = 10. Step 4: So T(t) = 3t² + 12t + 10. Step 5: To find R(t) = (f ÷ g)(t) = f(t)/g(t) = (3t² + 7t + 1)/(5t + 9). Step 6: The denominator cannot be zero, so set 5t + 9 = 0, which gives t = -9/5. Since t represents time in years, t >= 0. Therefore, the domain of R(t) in this ecological context is all t >= 0 (since -9/5 is negative and irrelevant). Step 7: The final answers are T(t) = 3t² + 12t + 10 and R(t) = (3t² + 7t + 1)/(5t + 9) for t >= 0.

2. Liam is analyzing the graph formed by combining two functions on a coordinate plane: f(x) = x^2 - 25 and g(x) = 2x + 10. The graph of h(x) = (f/g)(x) has a vertical asymptote and a hole. Determine the coordinates (x, y) of the hole in the graph of h(x). Answer: (-5, -5) Solution: Factor f(x) = x^2 - 25 = (x - 5)(x + 5). Factor g(x) = 2x + 10 = 2(x + 5). Write h(x) = [(x - 5)(x + 5)] / [2(x + 5)].
   Full step-by-step solution

   Step 1: Factor f(x) = x^2 - 25 = (x - 5)(x + 5). Step 2: Factor g(x) = 2x + 10 = 2(x + 5). Step 3: Write h(x) = [(x - 5)(x + 5)] / [2(x + 5)]. Step 4: The common factor (x + 5) cancels, leaving h(x) = (x - 5)/2 for x ≠ -5. Step 5: The hole occurs at x = -5. Step 6: Find the y-coordinate by substituting x = -5 into the simplified function: h(-5) = (-5 - 5)/2 = (-10)/2 = -5. Step 7: The hole is at (-5, -5).

3. Emma is a civil engineer designing a suspension bridge. The main cable's height above the bridge deck is modeled by the quadratic function h(x) = 0.0005x² - 0.5x + 150, where x is the horizontal distance in meters from the west anchor. The vertical support cables are attached at regular intervals, and the tension in a support cable at position x is given by the linear function T(x) = 0.15x + 20 (in kilonewtons). The total vertical load on the bridge deck at any point x is the product of the cable height and the tension in the support cable, given by L(x) = h(x) × T(x). Determine the function L(x) that models the total vertical load on the bridge deck. Answer: L(x) = 0.000075x³ - 0.055x² + 12.5x + 3000 Solution: We have h(x) = 0.0005x² - 0.5x + 150 and T(x) = 0.15x + 20. We need to find L(x) = h(x) × T(x).
   Full step-by-step solution

   Step 1: We have h(x) = 0.0005x² - 0.5x + 150 and T(x) = 0.15x + 20. We need to find L(x) = h(x) × T(x). Step 2: Multiply each term of h(x) by each term of T(x): (0.0005x²)(0.15x) = 0.000075x³ (0.0005x²)(20) = 0.01x² (-0.5x)(0.15x) = -0.075x² (-0.5x)(20) = -10x (150)(0.15x) = 22.5x (150)(20) = 3000 Step 3: Combine like terms: x³ term: 0.000075x³ x² terms: 0.01x² - 0.075x² = -0.065x² x terms: -10x + 22.5x = 12.5x constant: 3000 Step 4: Therefore, L(x) = 0.000075x³ - 0.065x² + 12.5x + 3000. The answer is L(x) = 0.000075x³ - 0.065x² + 12.5x + 3000.

4. 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.

5. f(x) = 3x³ - 7x² + 9, g(x) = 2x³ + 5x - 4. Find (f+g)(x) = ? Answer: 5x³ - 7x² + 5x + 5 Solution: Write the addition of the two functions: (f+g)(x) = (3x³ - 7x² + 9) + (2x³ + 5x - 4) Group like terms: (3x³ + 2x³) + (-7x²) + (5x) + (9 - 4) Add the x³ terms: 3x³ + 2x³ = 5x³ The x² term remains: -7x² The x term remains: 5x Add the constant terms: 9 - 4 = 5 Combine all terms: 5x³ - 7x² + 5x + 5…
   Full step-by-step solution

   Step 1: Write the addition of the two functions: (f+g)(x) = (3x³ - 7x² + 9) + (2x³ + 5x - 4) Step 2: Group like terms: (3x³ + 2x³) + (-7x²) + (5x) + (9 - 4) Step 3: Add the x³ terms: 3x³ + 2x³ = 5x³ Step 4: The x² term remains: -7x² Step 5: The x term remains: 5x Step 6: Add the constant terms: 9 - 4 = 5 Step 7: Combine all terms: 5x³ - 7x² + 5x + 5 The answer is 5x³ - 7x² + 5x + 5.

6. f(x) = 2x³ - 5x² + 8, g(x) = x³ + 3x - 12. Find (f + g)(x) = ? Answer: 3x³ - 5x² + 3x - 4 Solution: Write out the functions: f(x) = 2x³ - 5x² + 8, g(x) = x³ + 3x - 12 Add the functions: (f + g)(x) = (2x³ - 5x² + 8) + (x³ + 3x - 12) Combine x³ terms: 2x³ + x³ = 3x³ Keep the x² term: -5x² Keep the x term: +3x Combine constant terms: 8 + (-12) = -4 Write the final result: 3x³ - 5x² + 3x - 4 The…
   Full step-by-step solution

   Step 1: Write out the functions: f(x) = 2x³ - 5x² + 8, g(x) = x³ + 3x - 12 Step 2: Add the functions: (f + g)(x) = (2x³ - 5x² + 8) + (x³ + 3x - 12) Step 3: Combine x³ terms: 2x³ + x³ = 3x³ Step 4: Keep the x² term: -5x² Step 5: Keep the x term: +3x Step 6: Combine constant terms: 8 + (-12) = -4 Step 7: Write the final result: 3x³ - 5x² + 3x - 4 The answer is 3x³ - 5x² + 3x - 4.