Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 1

1. Emma is analyzing the motion of a particle in a physics experiment. The particle's position is given by f(x) = sqrt(x - 4), where x represents time in seconds. She applies a transformation g(x) = 1/(x - 2) to model how the particle's energy changes with position. To understand how energy depends on time, Emma needs to find the domain of the composite function (g∘f)(x). What is the domain of this composite function? Answer: x > 8 Solution: When finding the domain of composite functions, you need to ensure that the input to the inner function is valid, and that the output of the inner function is a valid input for the outer function.
   Full step-by-step solution

   When finding the domain of composite functions, you need to ensure that the input to the inner function is valid, and that the output of the inner function is a valid input for the outer function. For square root functions, the radicand must be greater than or equal to zero. For rational functions, the denominator cannot equal zero. These constraints must be satisfied simultaneously for the composite function to be defined.

2. Aroha is a geotechnical engineer analyzing the stability of a rock formation. The depth D(t) of a borehole in meters, t hours after drilling begins, is modeled by D(t) = sqrt(2t - 16). The pressure P(D) in kilopascals at depth D is given by P(D) = 1 / (D - 13). To ensure the drilling equipment is safe, Aroha needs to find the domain of the composite function P(D(t)) that describes pressure as a function of time. What is the domain of P∘D? Answer: t >= 8 and t != 92.5, or in interval notation [8, 92.5) ∪ (92.5, ∞) Solution: Identify the inner function D(t) = sqrt(2t - 16) and the outer function P(D) = 1/(D - 13). For the inner function D(t), the radicand must be non-negative: 2t - 16 >= 0 => 2t >= 16 => t >= 8.
   Full step-by-step solution

   Step 1: Identify the inner function D(t) = sqrt(2t - 16) and the outer function P(D) = 1/(D - 13). Step 2: For the inner function D(t), the radicand must be non-negative: 2t - 16 >= 0 => 2t >= 16 => t >= 8. Step 3: For the outer function P(D), the denominator cannot be zero: D - 13 != 0 => D != 13. Step 4: For the composite function P(D(t)), we need D(t) != 13. Set sqrt(2t - 16) = 13. Step 5: Square both sides: 2t - 16 = 169 => 2t = 185 => t = 185/2 = 92.5. Step 6: Therefore, t cannot equal 92.5. Step 7: Combine the restrictions: t >= 8 from the inner function, and t != 92.5 from the outer function. Step 8: In interval notation, the domain is [8, 92.5) ∪ (92.5, ∞). Step 9: Since t represents time in hours, the domain makes sense: drilling begins at t = 8 hours (when the depth becomes defined), and there is a specific time (92.5 hours) when the pressure function is undefined due to division by zero. Answer: t >= 8 and t != 92.5, or in interval notation [8, 92.5) ∪ (92.5, ∞).

3. A marine biologist is studying the temperature-dependent growth of a specific algae species in a controlled environment. The water temperature T in degrees Celsius is modeled by T(t) = √(t + 4), where t is time in days. The algae growth rate G is given by G(T) = 1/(T - 2), which represents how quickly the algae population increases per day at a given temperature. To understand how the growth rate changes over time, the biologist needs to find the domain of the composite function G(T(t)). What is the domain of G∘T? Answer: t > 0 Solution: Step 1: Identify the domain of T(t) = √(t + 4) For a square root to be defined, the expression inside must be ≥ 0: t + 4 ≥ 0 t ≥ -4 Step 2: Identify the domain of G(T) = 1/(T - 2) For a rational function, the denominator cannot be zero: T - 2 ≠ 0 T ≠ 2 Step 3: Find where T(t) = 2 √(t + 4) = 2…
   Full step-by-step solution

   Step 1: Identify the domain of T(t) = √(t + 4) For a square root to be defined, the expression inside must be ≥ 0: t + 4 ≥ 0 t ≥ -4 Step 2: Identify the domain of G(T) = 1/(T - 2) For a rational function, the denominator cannot be zero: T - 2 ≠ 0 T ≠ 2 Step 3: Find where T(t) = 2 √(t + 4) = 2 Square both sides: t + 4 = 4 t = 0 Step 4: Combine the restrictions From T(t): t ≥ -4 From G(T): T(t) ≠ 2, which means t ≠ 0 Step 5: Determine the final domain We need t ≥ -4 AND t ≠ 0 However, in the real-world context of time, t represents days, so t must be positive. Therefore, the domain is t > 0. The answer is t > 0.

4. 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^(0.08t). The water quality in the bay is measured by a function Q(p) = 100/(1 + 0.002p), where p is the dolphin population. To understand how water quality changes over time, the biologist needs to find the composite function Q(P(t)) and determine its domain. What is the domain of Q(P(t))? Answer: t ≥ 0 Solution: P(t) = 500 * e^(0.08t) where t is time in years. Q(p) = 100 / (1 + 0.002p) where p is the dolphin population.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the given functions** We have: - Dolphin population: P(t) = 500 * e^(0.08t) where t is time in years. - Water quality: Q(p) = 100 / (1 + 0.002p) where p is the dolphin population. --- **Step 2: Find the composite function Q(P(t))** Substitute p = P(t) into Q(p): Q(P(t)) = 100 / [1 + 0.002 * (500 * e^(0.08t))] Simplify inside the denominator: 0.002 * 500 = 1 So: Q(P(t)) = 100 / [1 + 1 * e^(0.08t)] Q(P(t)) = 100 / [1 + e^(0.08t)] --- **Step 3: Determine the domain of Q(P(t))** Domain means all possible t values for which Q(P(t)) is defined. First, check domain of P(t): P(t) = 500 * e^(0.08t) is defined for all real t. But t is time in years since the study started, so realistically t ≥ 0. Mathematically, P(t) is defined for any t. Second, check domain of Q(p): Q(p) = 100 / (1 + 0.002p) This is undefined when denominator = 0: 1 + 0.002p = 0 0.002p = -1 p = -500 So Q(p) is undefined only if p = -500. --- **Step 4: See if P(t) can be -500** P(t) = 500 * e^(0.08t) e^(0.08t) is always positive for any real t. So P(t) > 0 for all t. Thus P(t) never equals -500. Therefore, Q(P(t)) is defined for all real t mathematically. --- **Step 5: Apply the real-world context** The problem says: "A marine biologist is studying the population growth..." Time t = 0 is the start of the study, so t cannot be negative in this context. Thus the practical domain is t ≥ 0. --- **Final Answer:** The domain of Q(P(t)) is t ≥ 0.

5. 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, ∞).

6. Ava is studying the decay of a radioactive isotope in a laboratory. The amount of the isotope remaining after t hours is modeled by A(t) = sqrt(t - 7), measured in grams. The radiation level R in millisieverts is given by R(A) = ln(A - 10), where A is the amount in grams. To understand how the radiation level changes over time, Ava needs to find the domain of the composite function R(A(t)). What is the domain of R∘A? Answer: t > 107 Solution: Identify the inner function A(t) = sqrt(t - 7) and the outer function R(A) = ln(A - 10). For the inner function A(t) = sqrt(t - 7) to be defined, the radicand must be non-negative: t - 7 >= 0, so t >= 7.
   Full step-by-step solution

   Step 1: Identify the inner function A(t) = sqrt(t - 7) and the outer function R(A) = ln(A - 10). Step 2: For the inner function A(t) = sqrt(t - 7) to be defined, the radicand must be non-negative: t - 7 >= 0, so t >= 7. Step 3: For the outer function R(A) = ln(A - 10) to be defined, the argument must be positive: A - 10 > 0, so A > 10. Step 4: Since the output of the inner function becomes the input to the outer function, we need A(t) > 10. Step 5: Substitute A(t) into the inequality: sqrt(t - 7) > 10. Step 6: Square both sides (both sides are non-negative): t - 7 > 100. Step 7: Solve for t: t > 107. Step 8: Combine with the restriction from Step 2 (t >= 7). Since t > 107 is more restrictive, the domain is t > 107. Step 9: In interval notation, the domain is (107, infinity). The answer is t > 107.