Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 2

1. A composite function h(x) = f(g(x)) is defined where f(x) = √(x - 3) and g(x) = 1/(x - 5). On a coordinate plane, the graph of g(x) shows a vertical asymptote at x = 5, with the curve approaching positive and negative infinity on either side. The graph of f(x) begins at point (3, 0) and increases as a square root curve to the right. Determine the domain of h(x) in interval notation by analyzing the visual constraints from both functions. Answer: (-∞, 5) ∪ (5, 16/3] Solution: Identify the domain of g(x) = 1/(x - 5). Since division by zero is undefined, x ≠ 5. So domain of g(x) is (-∞, 5) ∪ (5, ∞).
   Full step-by-step solution

   Step 1: Identify the domain of g(x) = 1/(x - 5). Since division by zero is undefined, x ≠ 5. So domain of g(x) is (-∞, 5) ∪ (5, ∞). Step 2: Identify the domain of f(x) = √(x - 3). Since square roots require non-negative inputs, x - 3 ≥ 0, so x ≥ 3. Domain of f(x) is [3, ∞). Step 3: For the composite function h(x) = f(g(x)), the output of g(x) must be in the domain of f(x). So we need g(x) ≥ 3. Step 4: Solve 1/(x - 5) ≥ 3. Case 1: When x - 5 > 0 (x > 5): 1/(x - 5) ≥ 3 1 ≥ 3(x - 5) 1 ≥ 3x - 15 16 ≥ 3x x ≤ 16/3 Since x > 5 and x ≤ 16/3, we get 5 < x ≤ 16/3 Case 2: When x - 5 < 0 (x < 5): 1/(x - 5) ≥ 3 Since the left side is negative (because denominator is negative) and 3 is positive, this inequality is never true. Step 5: Combine with the domain restriction from g(x): x ≠ 5. The domain is 5 < x ≤ 16/3, which in interval notation is (5, 16/3]. The answer is (5, 16/3].

2. Given f(x) = √(x + 3) and g(x) = 1/(x - 2), find the domain of (g∘f)(x) Answer: [ -3, 1 ) ∪ ( 1, ∞ ) Solution: Find the domain of f(x) = √(x + 3) The expression under the square root must be ≥ 0 x + 3 ≥ 0 x ≥ -3 So domain of f is [-3, ∞) Find the domain of g(x) = 1/(x - 2) The denominator cannot be 0 x - 2 ≠ 0 x ≠ 2 So domain of g is (-∞, 2) ∪ (2, ∞) For (g∘f)(x) = g(f(x)) = 1/(√(x + 3) - 2) We need…
   Full step-by-step solution

   Step 1: Find the domain of f(x) = √(x + 3) The expression under the square root must be ≥ 0 x + 3 ≥ 0 x ≥ -3 So domain of f is [-3, ∞) Step 2: Find the domain of g(x) = 1/(x - 2) The denominator cannot be 0 x - 2 ≠ 0 x ≠ 2 So domain of g is (-∞, 2) ∪ (2, ∞) Step 3: For (g∘f)(x) = g(f(x)) = 1/(√(x + 3) - 2) We need both: f(x) is defined AND g(f(x)) is defined Step 4: f(x) is defined when x ≥ -3 Step 5: g(f(x)) is defined when f(x) ≠ 2 √(x + 3) ≠ 2 x + 3 ≠ 4 x ≠ 1 Step 6: Combine both conditions x ≥ -3 AND x ≠ 1 Step 7: Write in interval notation [-3, 1) ∪ (1, ∞) The domain is [-3, 1) ∪ (1, ∞)

3. Given f(x) = √(x + 3) and g(x) = 1/(x - 2), find the domain of (g∘f)(x) in interval notation. Answer: [ -3, 1 ) ∪ ( 1, ∞ ) Solution: Find (g∘f)(x) = g(f(x)) = 1/(√(x + 3) - 2) Domain of f(x) = √(x + 3) requires x + 3 ≥ 0, so x ≥ -3 Domain of g(f(x)) requires the denominator √(x + 3) - 2 ≠ 0 Solve √(x + 3) - 2 = 0 → √(x + 3) = 2 → x + 3 = 4 → x = 1 Exclude x = 1 from the domain Combine restrictions: x ≥ -3 and x ≠ 1 Write in…
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = 1/(√(x + 3) - 2) Step 2: Domain of f(x) = √(x + 3) requires x + 3 ≥ 0, so x ≥ -3 Step 3: Domain of g(f(x)) requires the denominator √(x + 3) - 2 ≠ 0 Step 4: Solve √(x + 3) - 2 = 0 → √(x + 3) = 2 → x + 3 = 4 → x = 1 Step 5: Exclude x = 1 from the domain Step 6: Combine restrictions: x ≥ -3 and x ≠ 1 Step 7: Write in interval notation: [-3, 1) ∪ (1, ∞)

4. A marine biologist is studying the temperature-dependent growth of algae in a research tank. 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 G∘T? Answer: t > 0 Solution: 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 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 Since T(t) = √(t + 4), we…
   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 restriction from G to the output of T Since T(t) = √(t + 4), we need √(t + 4) ≠ 2 Square both sides: t + 4 ≠ 4, so t ≠ 0 Step 4: Combine all restrictions From T(t): t ≥ -4 From G(T): t ≠ 0 So the domain is t ≥ -4 AND t ≠ 0 Step 5: Consider the real-world context Since t represents time in hours, we need t ≥ 0 Combining with our mathematical restrictions: t ≥ 0 AND t ≠ 0, which simplifies to t > 0 The domain of G∘T is t > 0.

5. Given f(x) = √(x - 9) and g(x) = 1/(x² - 16), find the domain of (g∘f)(x) in interval notation. Answer: [9, 25) ∪ (25, ∞) Solution: Find (g∘f)(x) = g(f(x)) = g(√(x - 9)) = 1/((√(x - 9))² - 16) = 1/((x - 9) - 16) = 1/(x - 25). Domain restrictions from f(x) = √(x - 9): x - 9 ≥ 0 → x ≥ 9.
   Full step-by-step solution

   Step 1: Find (g∘f)(x) = g(f(x)) = g(√(x - 9)) = 1/((√(x - 9))² - 16) = 1/((x - 9) - 16) = 1/(x - 25). Step 2: Domain restrictions from f(x) = √(x - 9): x - 9 ≥ 0 → x ≥ 9. Step 3: Domain restrictions from g(f(x)) = 1/(x - 25): x - 25 ≠ 0 → x ≠ 25. Step 4: Combine restrictions: x ≥ 9 AND x ≠ 25. Step 5: Write in interval notation: [9, 25) ∪ (25, ∞). The domain is [9, 25) ∪ (25, ∞).

6. Liam is designing a custom lens for a telescope that follows the path of a comet. The comet's position is modeled by the function f(x) = √(x - 2), and the telescope's magnification is given by g(x) = 3/(x - 5). Liam needs to determine the domain of the composite function (g∘f)(x) that represents the telescope's view of the comet. What is the domain of (g∘f)(x)? Answer: (2, 9) ∪ (9, ∞) Solution: Composite function domains require analyzing restrictions step by step.
   Full step-by-step solution

   Composite function domains require analyzing restrictions step by step. First, the inner function must be defined, which often involves restrictions like non-negative values under square roots or denominators not equaling zero. Then, the output of the inner function must be valid input for the outer function, which may introduce additional restrictions. The final domain is the intersection of all these conditions, excluding any values that make any part of the composition undefined.

7. Liam is analyzing the temperature fluctuations in a chemical reaction. The temperature T in degrees Celsius is modeled by T(t) = √(t - 2), where t is time in hours. The reaction rate R is given by R(T) = 1/(T - 3). Liam needs to find the domain of the composite function R(T(t)), which represents how the reaction rate depends on time. What is the domain of R∘T? Answer: (2, 7) ∪ (7, ∞) Solution: Composite functions require analyzing domains step by step. First, determine the domain of the inner function by identifying where its expression is defined.
   Full step-by-step solution

   Composite functions require analyzing domains step by step. First, determine the domain of the inner function by identifying where its expression is defined. Then, consider how the output of the inner function affects the domain of the outer function. The final domain consists of all input values where both functions are defined and the composition makes mathematical sense. This concept is important in mathematical modeling where multiple relationships are chained together.