Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 2

1. Charlotte is a pharmacist formulating a slow-release medication. The concentration of the drug in a patient's bloodstream t hours after administration is modeled by C(t) = ln(t - 9), where C is in milligrams per liter. The body's response rate R to the drug concentration is given by R(C) = 1 / (C - 12), measured in arbitrary units. To ensure patient safety, Charlotte needs to determine the domain of the composite function R(C(t)) that describes the response rate as a function of time. What is the domain of R∘C? Answer: t > 9 and t ≠ 9 + e^12 Solution: Identify the inner function: C(t) = ln(t - 9). For the natural logarithm to be defined, its argument must be positive: t - 9 > 0, so t > 9. This gives the first restriction.
   Full step-by-step solution

   Step 1: Identify the inner function: C(t) = ln(t - 9). For the natural logarithm to be defined, its argument must be positive: t - 9 > 0, so t > 9. This gives the first restriction. Step 2: Identify the outer function: R(C) = 1/(C - 12). For this rational function to be defined, the denominator cannot be zero: C - 12 ≠ 0, so C ≠ 12. Step 3: For the composite function R(C(t)), we need the output of C(t) to be a valid input for R. This means C(t) ≠ 12. So we solve ln(t - 9) ≠ 12. Step 4: Solve ln(t - 9) = 12. Exponentiate both sides: e^(ln(t - 9)) = e^12, so t - 9 = e^12, thus t = 9 + e^12. Step 5: The value t = 9 + e^12 makes C(t) = 12, which makes the denominator zero in R, so it must be excluded. Step 6: Combine the restrictions: t must satisfy t > 9 (from the logarithm) and t ≠ 9 + e^12 (from the denominator). Step 7: Therefore, the domain is all real numbers t such that t > 9 and t ≠ 9 + e^12. In interval notation: (9, 9 + e^12) ∪ (9 + e^12, ∞).

2. Sophia is modeling the spread of a beneficial fungus through a forest's root system. The area A(t) in square meters covered by the fungus after t weeks is given by A(t) = ln(t - 7). The nutrient absorption rate N (in grams per week) depends on the area covered and is modeled by N(A) = 1 / (A - 9). Sophia wants to understand how the nutrient absorption rate changes over time by studying the composite function N(A(t)). Determine the domain of this composite function, considering the real-world context that time t is measured in weeks. Answer: t > 7 and t ≠ 7 + e^9 Solution: Identify the inner function: A(t) = ln(t - 7). For the natural logarithm to be defined, its argument must be positive: t - 7 > 0, so t > 7. Identify the outer function: N(A) = 1 / (A - 9).
   Full step-by-step solution

   Step 1: Identify the inner function: A(t) = ln(t - 7). For the natural logarithm to be defined, its argument must be positive: t - 7 > 0, so t > 7. Step 2: Identify the outer function: N(A) = 1 / (A - 9). For this rational function to be defined, the denominator cannot be zero: A - 9 ≠ 0, so A ≠ 9. Step 3: For the composite function N(A(t)) to be defined, we need both: - t > 7 (from step 1) - A(t) ≠ 9 (from step 2) Step 4: Find when A(t) = 9. Solve ln(t - 7) = 9. Exponentiate both sides: t - 7 = e^9, so t = 7 + e^9. Step 5: Combine the restrictions. The domain is all t > 7 except t = 7 + e^9. Step 6: In interval notation: (7, 7 + e^9) ∪ (7 + e^9, ∞). Therefore, the domain of N(A(t)) is t > 7 and t ≠ 7 + e^9.

3. A marine biologist is studying the population growth of dolphins in a protected bay. The dolphin population P(t) after t years is modeled by P(t) = 500e^(0.03t). The available food supply F in kilograms is given by F(p) = 2000 - 2p, where p is the dolphin population. Determine the domain of the composite function F(P(t)) that represents the food supply as a function of time, considering the biological constraints that both population and food supply must remain non-negative. Answer: [0, 1000/3] or approximately [0, 333.33] years Solution: For exponential growth models combined with resource constraints, we typically find where the inner function's output exceeds the outer function's domain limits.
   Full step-by-step solution

   In function composition problems with real-world applications, we need to ensure that the output of the inner function falls within the domain of the outer function. For exponential growth models combined with resource constraints, we typically find where the inner function's output exceeds the outer function's domain limits. This often involves solving inequalities to determine the valid time range where all conditions are satisfied.

4. 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) = sqrt(t - 5), where t is time in days. The coral growth rate G is given by G(T) = 1/(T - 3). To understand how growth rate depends on time, the biologist needs to find the domain of the composite function G(T(t)). What is the domain of G∘T? Answer: t > 14 Solution: Step 1: Identify the inner function T(t) = sqrt(t - 5) For T(t) to be defined, the expression inside the square root must be non-negative: t - 5 ≥ 0 t ≥ 5 Step 2: Identify the outer function G(T) = 1/(T - 3) For G(T) to be defined, the denominator cannot be zero: T - 3 ≠ 0 T ≠ 3 Step 3:…
   Full step-by-step solution

   Step 1: Identify the inner function T(t) = sqrt(t - 5) For T(t) to be defined, the expression inside the square root must be non-negative: t - 5 ≥ 0 t ≥ 5 Step 2: Identify the outer function G(T) = 1/(T - 3) For G(T) to be defined, the denominator cannot be zero: T - 3 ≠ 0 T ≠ 3 Step 3: Substitute T(t) into the restriction for G Since T(t) = sqrt(t - 5), we need: sqrt(t - 5) ≠ 3 Step 4: Solve the inequality Square both sides: t - 5 ≠ 9 t ≠ 14 Step 5: Combine all restrictions From Step 1: t ≥ 5 From Step 4: t ≠ 14 Therefore, the domain is t ≥ 5 AND t ≠ 14 Step 6: Verify the restriction t ≠ 14 When t = 14, T(14) = sqrt(14 - 5) = sqrt(9) = 3 Then G(T(14)) = 1/(3 - 3) = 1/0, which is undefined Step 7: Write the final domain in interval notation The domain is [5, 14) ∪ (14, ∞) In inequality form: t > 14 or 5 ≤ t < 14 Since the question asks for the domain of G∘T, and t must satisfy both function definitions, the answer is t > 14.

5. A marine biologist is studying the temperature-dependent growth of algae in a research aquarium. The water temperature T in degrees Celsius is modeled by T(t) = √(t + 4), where t is time in hours. The algae growth rate G is given by G(T) = 1/(T - 2), measured in cells per hour. To understand how the algae growth rate changes 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: Step 1: Identify the inner function T(t) = √(t + 4) For T(t) to be defined, the expression under the square root must be non-negative: t + 4 ≥ 0, so t ≥ -4 Step 2: Identify the outer function G(T) = 1/(T - 2) For G(T) to be defined, the denominator cannot be zero: T - 2 ≠ 0, so T ≠ 2 Step 3:…
   Full step-by-step solution

   Step 1: Identify the inner function T(t) = √(t + 4) For T(t) to be defined, the expression under the square root must be non-negative: t + 4 ≥ 0, so t ≥ -4 Step 2: Identify the outer function G(T) = 1/(T - 2) For G(T) to be defined, the denominator cannot be zero: T - 2 ≠ 0, so T ≠ 2 Step 3: Apply the outer function's restriction to the inner function We need T(t) ≠ 2, which means √(t + 4) ≠ 2 Square both sides: t + 4 ≠ 4, so t ≠ 0 Step 4: Combine all restrictions From Step 1: t ≥ -4 From Step 3: t ≠ 0 Also, in the real-world context, time t must be positive: t > 0 Step 5: Final domain The domain is t > 0 (since t ≥ -4 and t ≠ 0 and t > 0 simplifies to t > 0)

6. Given f(x) = ln(3x - 6) and g(x) = sqrt(x^2 - 16), find the number of integers in the domain of the composite function (f∘g)(x). Answer: 4 Solution: Find the domain of g(x) = sqrt(x^2 - 16) For square root functions, the expression inside must be ≥ 0 x^2 - 16 ≥ 0 x^2 ≥ 16 |x| ≥ 4 So x ≤ -4 or x ≥ 4 Find the domain of f(x) = ln(3x - 6) For logarithmic functions, the expression inside must be > 0 3x - 6 > 0 3x > 6 x > 2 For (f∘g)(x) = f(g(x)),…
   Full step-by-step solution

   Step 1: Find the domain of g(x) = sqrt(x^2 - 16) For square root functions, the expression inside must be ≥ 0 x^2 - 16 ≥ 0 x^2 ≥ 16 |x| ≥ 4 So x ≤ -4 or x ≥ 4 Step 2: Find the domain of f(x) = ln(3x - 6) For logarithmic functions, the expression inside must be > 0 3x - 6 > 0 3x > 6 x > 2 Step 3: For (f∘g)(x) = f(g(x)), we need g(x) to be in the domain of f This means g(x) > 2 Since g(x) = sqrt(x^2 - 16) ≥ 0, we need sqrt(x^2 - 16) > 2 Step 4: Solve sqrt(x^2 - 16) > 2 Square both sides: x^2 - 16 > 4 x^2 > 20 |x| > sqrt(20) |x| > 2sqrt(5) Since sqrt(5) ≈ 2.236, 2sqrt(5) ≈ 4.472 So x < -4.472 or x > 4.472 Step 5: Combine with domain of g(x): x ≤ -4 or x ≥ 4 Intersection gives: x < -4.472 or x > 4.472 Step 6: Find the integers in this domain x < -4.472 means x ≤ -5 x > 4.472 means x ≥ 5 So the integers are: ..., -7, -6, -5, 5, 6, 7, ... Step 7: Count the integers There are infinitely many integers in the domain, so the number of integers in the domain is infinite.