Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 3

1. 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

2. Given f(x) = √(x - 6) and g(x) = 1/(x² - 121), find the domain of (g∘f)(x) in interval notation. Answer: [6, 17) ∪ (17, ∞) Solution: Find (g∘f)(x) = g(f(x)) = g(√(x - 6)) = 1/((√(x - 6))² - 121) = 1/((x - 6) - 121) = 1/(x - 127). Domain restrictions from f(x) = √(x - 6): x - 6 ≥ 0 → x ≥ 6.
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = g(√(x - 6)) = 1/((√(x - 6))² - 121) = 1/((x - 6) - 121) = 1/(x - 127). Step 2: Domain restrictions from f(x) = √(x - 6): x - 6 ≥ 0 → x ≥ 6. Step 3: Domain restrictions from g(f(x)) = 1/(x - 127): x - 127 ≠ 0 → x ≠ 127. Step 4: Also, the output of f(x) must be in the domain of g(x). g(x) = 1/(x² - 121) requires x² - 121 ≠ 0, so x ≠ ±11. Since f(x) = √(x - 6) ≥ 0, we need √(x - 6) ≠ 11 → x - 6 ≠ 121 → x ≠ 127 (already covered). Also √(x - 6) ≠ -11 is impossible since sqrt is non-negative. So no new restriction from the ±11 condition. Step 5: Combine restrictions: x ≥ 6 AND x ≠ 127. Step 6: Write in interval notation: [6, 127) ∪ (127, ∞). The domain is [6, 127) ∪ (127, ∞).

3. Given f(x) = √(x - 12) and g(x) = 1/(x² - 81), find the domain of (g∘f)(x) in interval notation. Answer: [12, 21) ∪ (21, ∞) Solution: Find (g∘f)(x) = g(f(x)) = g(√(x - 12)) = 1/((√(x - 12))² - 81) = 1/((x - 12) - 81) = 1/(x - 93). Domain restrictions from f(x) = √(x - 12): x - 12 ≥ 0 → x ≥ 12.
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = g(√(x - 12)) = 1/((√(x - 12))² - 81) = 1/((x - 12) - 81) = 1/(x - 93). Step 2: Domain restrictions from f(x) = √(x - 12): x - 12 ≥ 0 → x ≥ 12. Step 3: Domain restrictions from g(f(x)) = 1/(x - 93): x - 93 ≠ 0 → x ≠ 93. Step 4: Also, the output of f(x) must be in the domain of g(x). g(x) = 1/(x² - 81) requires x² - 81 ≠ 0, so x ≠ ±9. Since f(x) = √(x - 12) ≥ 0, we need √(x - 12) ≠ 9 → x - 12 ≠ 81 → x ≠ 93 (already covered). Also √(x - 12) ≠ -9 is impossible since sqrt is non-negative. So no new restriction from the ±9 condition. Step 5: Combine restrictions: x ≥ 12 AND x ≠ 93. Step 6: Write in interval notation: [12, 93) ∪ (93, ∞). The domain is [12, 93) ∪ (93, ∞).

4. f(x) = √(x + 3) and g(x) = 1/(x - 2), find the domain of (f ∘ g)(x) Answer: (-∞, 2) ∪ (2, ∞) Solution: Write the composition: (f ∘ g)(x) = f(g(x)) = √(1/(x - 2) + 3) For g(x) to be defined: x - 2 ≠ 0, so x ≠ 2 For f(g(x)) to be defined: g(x) + 3 ≥ 0 Solve: 1/(x - 2) + 3 ≥ 0 Find common denominator: (1 + 3(x - 2))/(x - 2) ≥ 0 Simplify: (3x - 5)/(x - 2) ≥ 0 Critical points: x = 5/3 and x = 2 Test…
   Full step-by-step solution

   Step 1: Write the composition: (f ∘ g)(x) = f(g(x)) = √(1/(x - 2) + 3) Step 2: For g(x) to be defined: x - 2 ≠ 0, so x ≠ 2 Step 3: For f(g(x)) to be defined: g(x) + 3 ≥ 0 Step 4: Solve: 1/(x - 2) + 3 ≥ 0 Step 5: Find common denominator: (1 + 3(x - 2))/(x - 2) ≥ 0 Step 6: Simplify: (3x - 5)/(x - 2) ≥ 0 Step 7: Critical points: x = 5/3 and x = 2 Step 8: Test intervals: (-∞, 5/3), (5/3, 2), (2, ∞) Step 9: For x < 5/3: negative/negative = positive (≥ 0) Step 10: For 5/3 < x < 2: positive/negative = negative (< 0) Step 11: For x > 2: positive/positive = positive (≥ 0) Step 12: Combine with x ≠ 2: Domain is (-∞, 5/3] ∪ (2, ∞) Step 13: Check x = 5/3: (3(5/3) - 5)/(5/3 - 2) = (5 - 5)/(-1/3) = 0/(-1/3) = 0 ≥ 0 ✓ The domain is (-∞, 5/3] ∪ (2, ∞).

5. Given f(x) = ln(x - 3) and g(x) = √(x + 1), find the domain of (f∘g)(x) Answer: [4, ∞) Solution: Find the domain of g(x) = √(x + 1) The expression under the square root must be non-negative: x + 1 ≥ 0 → x ≥ -1 So domain of g is [-1, ∞) Find the domain of f(x) = ln(x - 3) The argument of the natural logarithm must be positive: x - 3 > 0 → x > 3 So domain of f is (3, ∞) For (f∘g)(x) = f(g(x))…
   Full step-by-step solution

   Step 1: Find the domain of g(x) = √(x + 1) The expression under the square root must be non-negative: x + 1 ≥ 0 → x ≥ -1 So domain of g is [-1, ∞) Step 2: Find the domain of f(x) = ln(x - 3) The argument of the natural logarithm must be positive: x - 3 > 0 → x > 3 So domain of f is (3, ∞) Step 3: For (f∘g)(x) = f(g(x)) to be defined: - g(x) must be defined → x ≥ -1 - g(x) must be in the domain of f → g(x) > 3 Step 4: Solve g(x) > 3 √(x + 1) > 3 Square both sides: x + 1 > 9 x > 8 Step 5: Combine conditions: x ≥ -1 AND x > 8 → x > 8 Step 6: Check the boundary: when x = 8, g(8) = √(8 + 1) = √9 = 3 But f(3) = ln(3 - 3) = ln(0) is undefined So x = 8 is NOT included Final answer: Domain is (8, ∞)

6. 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.

7. Liam is a biomedical engineer developing a new method to monitor drug concentration in a patient's bloodstream. The concentration C(t) of the drug in milligrams per liter, t hours after injection, is modeled by C(t) = ln(t - 9). The drug's effectiveness E(C), measured in arbitrary units, is given by E(C) = 1 / (C - 11). To understand how the drug's effectiveness changes over time, Liam needs to determine the domain of the composite function E(C(t)). What is the domain of E∘C? Answer: t > 9 and t ≠ 9 + e^11, or in interval notation (9, 9 + e^11) ∪ (9 + e^11, ∞) Solution: Identify the inner function C(t) = ln(t - 9) and the outer function E(C) = 1/(C - 11). For the inner function C(t), the argument of the natural logarithm must be positive: t - 9 > 0, so t > 9.
   Full step-by-step solution

   Step 1: Identify the inner function C(t) = ln(t - 9) and the outer function E(C) = 1/(C - 11). Step 2: For the inner function C(t), the argument of the natural logarithm must be positive: t - 9 > 0, so t > 9. Step 3: For the outer function E(C), the denominator cannot be zero: C - 11 ≠ 0, so C ≠ 11. Step 4: For the composite function E(C(t)), we need C(t) ≠ 11. Set ln(t - 9) = 11 and solve: t - 9 = e^11, so t = 9 + e^11. Step 5: Therefore, t cannot equal 9 + e^11. Step 6: Combine the restrictions: t > 9 from the inner function, and t ≠ 9 + e^11 from the outer function. Step 7: In interval notation, the domain is (9, 9 + e^11) ∪ (9 + e^11, ∞). Step 8: Since t represents time in hours, the domain makes sense in the real-world context: the drug begins to have an effect after 9 hours (when the concentration becomes defined), and there is a specific time (9 + e^11 hours) when the effectiveness function is undefined due to division by zero. Answer: t > 9 and t ≠ 9 + e^11, or in interval notation (9, 9 + e^11) ∪ (9 + e^11, ∞).