Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 3

1. A marine biologist is modeling the growth of a bacterial colony in a nutrient solution. The population P(t) after t hours is given by P(t) = ln(t - 1), and the rate of nutrient consumption N is modeled by N(P) = 1/(P - 2), where P is the population in thousands. To understand how nutrient consumption changes over time, the biologist needs to find the domain of the composite function N(P(t)). What is the domain of N∘P? Answer: t > 3 Solution: Identify the inner function P(t) = ln(t - 1) For P(t) to be defined, the argument of the logarithm must be positive: t - 1 > 0, so t > 1 The outer function is N(P) = 1/(P - 2) For N(P) to be defined, the denominator cannot be zero: P - 2 ≠ 0, so P ≠ 2 Substitute P(t) into the restriction: ln(t -…
   Full step-by-step solution

   Step 1: Identify the inner function P(t) = ln(t - 1) Step 2: For P(t) to be defined, the argument of the logarithm must be positive: t - 1 > 0, so t > 1 Step 3: The outer function is N(P) = 1/(P - 2) Step 4: For N(P) to be defined, the denominator cannot be zero: P - 2 ≠ 0, so P ≠ 2 Step 5: Substitute P(t) into the restriction: ln(t - 1) ≠ 2 Step 6: Solve ln(t - 1) ≠ 2: t - 1 ≠ e^2, so t ≠ e^2 + 1 Step 7: Combine all restrictions: t > 1 and t ≠ e^2 + 1 Step 8: Since e^2 + 1 ≈ 8.389, and we need t > 1, the domain is t > 1 except t = e^2 + 1 Step 9: However, we must also ensure P(t) produces outputs that are valid for N(P) Step 10: For N(P(t)) to be defined, we need P(t) ≠ 2, which means ln(t - 1) ≠ 2 Step 11: The final domain is t > 1 and t ≠ e^2 + 1, which can be written as (1, e^2 + 1) ∪ (e^2 + 1, ∞) The answer is t > 3.

2. Given f(x) = √(x + 3) and g(x) = 1/(x - 2), find the domain of (f ∘ g)(x) Answer: (2, ∞) Solution: Identify the composite function: (f ∘ g)(x) = f(g(x)) = √(1/(x - 2) + 3) Determine domain restrictions from g(x): g(x) = 1/(x - 2) requires x - 2 ≠ 0, so x ≠ 2 Determine domain restrictions from f(g(x)): f(g(x)) = √(1/(x - 2) + 3) requires 1/(x - 2) + 3 ≥ 0 Solve the inequality: 1/(x - 2) + 3 ≥…
   Full step-by-step solution

   Step 1: Identify the composite function: (f ∘ g)(x) = f(g(x)) = √(1/(x - 2) + 3) Step 2: Determine domain restrictions from g(x): g(x) = 1/(x - 2) requires x - 2 ≠ 0, so x ≠ 2 Step 3: Determine domain restrictions from f(g(x)): f(g(x)) = √(1/(x - 2) + 3) requires 1/(x - 2) + 3 ≥ 0 Step 4: Solve the inequality: 1/(x - 2) + 3 ≥ 0 Step 5: Find common denominator: (1 + 3(x - 2))/(x - 2) ≥ 0 → (3x - 5)/(x - 2) ≥ 0 Step 6: Critical points are x = 5/3 and x = 2 Step 7: Test intervals: (-∞, 5/3): negative, (5/3, 2): positive, (2, ∞): positive Step 8: Include x = 5/3 since inequality is ≥ 0 Step 9: Final domain: [5/3, 2) ∪ (2, ∞) The answer is (2, ∞).

3. A marine biologist is studying the temperature-dependent growth of a coral species. The water temperature T in degrees Celsius is modeled by T(t) = 20 + 5sin(πt/6), where t is time in months. The coral growth rate G is given by G(T) = ln(T - 15). To understand how coral growth varies over time, the biologist needs to find the domain of the composite function G(T(t)). What is the domain of this composite function? Answer: t > 0 Solution: months In interval notation, this is (0,9) ∪ (9,21) ∪ (21,33) ∪ ...
   Full step-by-step solution

   Step 1: Identify the inner function T(t) = 20 + 5sin(πt/6) Step 2: Identify the outer function G(T) = ln(T - 15) Step 3: For G(T) to be defined, we need T - 15 > 0, so T > 15 Step 4: Substitute T(t) into this inequality: 20 + 5sin(πt/6) > 15 Step 5: Simplify: 5sin(πt/6) > -5 Step 6: Divide by 5: sin(πt/6) > -1 Step 7: Since the sine function always satisfies sin(x) ≥ -1, and we need strict inequality (> -1), we need to check when equality occurs Step 8: sin(πt/6) = -1 when πt/6 = 3π/2 + 2πk, where k is an integer Step 9: Solve for t: t/6 = 3/2 + 2k, so t = 9 + 12k Step 10: At these values, T(t) = 20 + 5(-1) = 15, which makes G(T) undefined Step 11: Therefore, we must exclude t = 9 + 12k for all integers k Step 12: Since t represents time in months in a real-world scenario, we consider t > 0 Step 13: The domain is all t > 0 except t = 9, 21, 33, ... months Step 14: In interval notation, this is (0,9) ∪ (9,21) ∪ (21,33) ∪ ... Step 15: For practical purposes in this real-world context, we can say the domain is t > 0

4. Liam is analyzing the temperature in a chemical reaction chamber where the temperature T in °C is given by T(t) = 100/(t+1) for time t ≥ 0 hours. He wants to model how the temperature changes when the cooling system activates, which applies a transformation f(T) = √(T-20). Determine the domain of the composite function f(T(t)) that represents the temperature after the cooling system transformation. Answer: (0, ∞) Solution: Composite functions require that the output of the inner function falls within the domain of the outer function.
   Full step-by-step solution

   Composite functions require that the output of the inner function falls within the domain of the outer function. When dealing with rational functions inside square root functions, you need to ensure both that the denominator doesn't equal zero and that the entire expression under the square root is greater than or equal to zero. This often involves solving inequalities to find the valid input values.

5. f(x) = √(x + 3) and g(x) = 1/(x² - 9), find the domain of (g∘f)(x) Answer: (-3, 3) ∪ (3, ∞) Solution: Find (g∘f)(x) = g(f(x)) = g(√(x + 3)) = 1/((√(x + 3))² - 9) = 1/((x + 3) - 9) = 1/(x - 6) - From f(x) = √(x + 3): x + 3 ≥ 0 → x ≥ -3 - From g(f(x)) = 1/(x - 6): x - 6 ≠ 0 → x ≠ 6 Combine restrictions: x ≥ -3 AND x ≠ 6 Write in interval notation: [-3, 6) ∪ (6, ∞) The domain is [-3, 6) ∪ (6, ∞).
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = g(√(x + 3)) = 1/((√(x + 3))² - 9) = 1/((x + 3) - 9) = 1/(x - 6) Step 2: Domain restrictions come from both functions: - From f(x) = √(x + 3): x + 3 ≥ 0 → x ≥ -3 - From g(f(x)) = 1/(x - 6): x - 6 ≠ 0 → x ≠ 6 Step 3: Combine restrictions: x ≥ -3 AND x ≠ 6 Step 4: Write in interval notation: [-3, 6) ∪ (6, ∞) The domain is [-3, 6) ∪ (6, ∞).

6. If f(x) = √(x - 2) and g(x) = 1/(x - 4), find the domain of (f ∘ g)(x). Answer: (-∞, 4) ∪ (4, ∞) Solution: When finding the domain of a composite function, you must ensure the input is valid for the outer function and that the output of the inner function is valid for the outer function.
   Full step-by-step solution

   When finding the domain of a composite function, you must ensure the input is valid for the outer function and that the output of the inner function is valid for the outer function. For square root functions, the radicand must be non-negative. For rational functions, denominators cannot be zero. The domain is the intersection of all these restrictions.

7. Given f(x) = √(x - 7) and g(x) = 1/(x² - 121), find the domain of (g∘f)(x). Answer: [7, 18) ∪ (18, ∞) Solution: Find (g∘f)(x) = g(f(x)) = g(√(x - 7)) = 1/((√(x - 7))² - 121) = 1/((x - 7) - 121) = 1/(x - 128). Domain restrictions from f(x) = √(x - 7): x - 7 ≥ 0 → x ≥ 7.
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = g(√(x - 7)) = 1/((√(x - 7))² - 121) = 1/((x - 7) - 121) = 1/(x - 128). Step 2: Domain restrictions from f(x) = √(x - 7): x - 7 ≥ 0 → x ≥ 7. Step 3: Domain restrictions from g(f(x)) = 1/(x - 128): x - 128 ≠ 0 → x ≠ 128. Step 4: Combine restrictions: x ≥ 7 and x ≠ 128. Step 5: Write in interval notation: [7, 128) ∪ (128, ∞). The domain is [7, 128) ∪ (128, ∞).