Composite Functions

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Answer Key & Explanations

Composite Functions · Grade 12 · Worksheet 1

1. Given f(x) = √(x + 3) and g(x) = 1/(x - 1), find the domain of (f ∘ g)(x) Answer: (-∞, -2) ∪ (-2, 1) ∪ (1, ∞) Solution: Identify the composite function (f ∘ g)(x) = f(g(x)) = √(1/(x - 1) + 3) Determine domain restrictions from g(x): x - 1 ≠ 0, so x ≠ 1 Determine domain restrictions from f(g(x)): The expression inside the square root must be ≥ 0 Set up inequality: 1/(x - 1) + 3 ≥ 0 Find common denominator: (1 +…
   Full step-by-step solution

   Step 1: Identify the composite function (f ∘ g)(x) = f(g(x)) = √(1/(x - 1) + 3) Step 2: Determine domain restrictions from g(x): x - 1 ≠ 0, so x ≠ 1 Step 3: Determine domain restrictions from f(g(x)): The expression inside the square root must be ≥ 0 Step 4: Set up inequality: 1/(x - 1) + 3 ≥ 0 Step 5: Find common denominator: (1 + 3(x - 1))/(x - 1) ≥ 0 Step 6: Simplify numerator: (1 + 3x - 3)/(x - 1) ≥ 0 → (3x - 2)/(x - 1) ≥ 0 Step 7: Find critical points: 3x - 2 = 0 → x = 2/3, and x - 1 = 0 → x = 1 Step 8: Test intervals: (-∞, 2/3), (2/3, 1), (1, ∞) Step 9: For x < 2/3: (negative)/(negative) = positive ✓ Step 10: For 2/3 < x < 1: (positive)/(negative) = negative ✗ Step 11: For x > 1: (positive)/(positive) = positive ✓ Step 12: Include x = 2/3 since inequality is ≥ 0 Step 13: Combine all restrictions: x ≠ 1 and x ∈ (-∞, 2/3] ∪ (1, ∞) Step 14: Final domain: (-∞, 2/3] ∪ (1, ∞)

2. A marine biologist is studying the population dynamics of a fish species in a lake. The fish population P(t) after t years is modeled by P(t) = 1000e^(0.05t). The water quality function Q(p) = 1/(p - 800) measures how water quality depends on the fish population p. To understand how water quality changes over time, the biologist needs to find the domain of the composite function Q(P(t)). What is the domain of Q(P(t)) in this real-world context? Answer: t > ln(800/1000)/0.05 Solution: When finding domains of composite functions, you need to ensure the output of the inner function falls within the domain of the outer function.
   Full step-by-step solution

   When finding domains of composite functions, you need to ensure the output of the inner function falls within the domain of the outer function. For rational functions, denominators cannot be zero, and for real-world applications, values must make practical sense.

3. Given f(x) = √(x + 3) and g(x) = 1/(x² - 9), find the domain of (g∘f)(x) Answer: (-3, 3) ∪ (3, ∞) Solution: Step 1: Write the composition (g∘f)(x) = g(f(x)) = 1/((√(x + 3))² - 9) = 1/((x + 3) - 9) = 1/(x - 6) Step 2: Find domain restrictions from f(x) = √(x + 3) - The radicand must be non-negative: x + 3 ≥ 0 - Therefore: x ≥ -3 Step 3: Find domain restrictions from g(f(x)) = 1/(x - 6) - The…
   Full step-by-step solution

   Step 1: Write the composition (g∘f)(x) = g(f(x)) = 1/((√(x + 3))² - 9) = 1/((x + 3) - 9) = 1/(x - 6) Step 2: Find domain restrictions from f(x) = √(x + 3) - The radicand must be non-negative: x + 3 ≥ 0 - Therefore: x ≥ -3 Step 3: Find domain restrictions from g(f(x)) = 1/(x - 6) - The denominator cannot be zero: x - 6 ≠ 0 - Therefore: x ≠ 6 Step 4: Combine all restrictions - From Step 2: x ≥ -3 - From Step 3: x ≠ 6 - Therefore domain is: [-3, 6) ∪ (6, ∞) Step 5: Verify by checking if any values in [-3, 6) ∪ (6, ∞) cause issues - At x = -3: f(-3) = √(0) = 0, g(0) = 1/(0² - 9) = -1/9 ✓ - At x = 6: f(6) = √9 = 3, g(3) = 1/(9 - 9) = undefined ✗ - All other values in the domain work correctly Final answer: [-3, 6) ∪ (6, ∞)

4. Maria is analyzing the motion of a particle in a physics experiment. The particle's position is given by f(t) = sqrt(t - 4), where t is time in seconds. She applies a transformation function g(x) = 1/(x - 2) to model how the particle's energy changes with position. Determine the domain of the composite function (g∘f)(t) that represents the energy as a function of time. Answer: t > 8 Solution: The composite function is (g∘f)(t) = g(f(t)) = 1/(sqrt(t - 4) - 2) For f(t) = sqrt(t - 4) to be defined, the expression under the square root must be non-negative: t - 4 ≥ 0, so t ≥ 4 For g(f(t)) to be defined, the denominator cannot be zero: sqrt(t - 4) - 2 ≠ 0 Solve sqrt(t - 4) - 2 ≠ 0: sqrt(t…
   Full step-by-step solution

   Step 1: The composite function is (g∘f)(t) = g(f(t)) = 1/(sqrt(t - 4) - 2) Step 2: For f(t) = sqrt(t - 4) to be defined, the expression under the square root must be non-negative: t - 4 ≥ 0, so t ≥ 4 Step 3: For g(f(t)) to be defined, the denominator cannot be zero: sqrt(t - 4) - 2 ≠ 0 Step 4: Solve sqrt(t - 4) - 2 ≠ 0: sqrt(t - 4) ≠ 2 Step 5: Square both sides: t - 4 ≠ 4, so t ≠ 8 Step 6: Combine both conditions: t ≥ 4 AND t ≠ 8 Step 7: However, we must also ensure the square root is defined in the real numbers, which gives t ≥ 4 Step 8: The domain is all t such that t ≥ 4 and t ≠ 8, which can be written as t > 8 or 4 ≤ t < 8 Step 9: In interval notation, this is [4, 8) ∪ (8, ∞) The answer is t > 8 or 4 ≤ t < 8.

5. Given f(x) = 1/(x - 3) and g(x) = √(x + 2), find the domain of (f ∘ g)(x) Answer: [ -2, 3 ) ∪ ( 3, ∞ ) Solution: Identify the composite function: (f ∘ g)(x) = f(g(x)) = 1/(√(x + 2) - 3) Find domain of g(x): √(x + 2) requires x + 2 ≥ 0, so x ≥ -2 Find domain of f(g(x)): f(g(x)) = 1/(√(x + 2) - 3) requires √(x + 2) - 3 ≠ 0 Solve √(x + 2) - 3 ≠ 0: √(x + 2) ≠ 3, x + 2 ≠ 9, x ≠ 7 Combine restrictions: x ≥ -2…
   Full step-by-step solution

   Step 1: Identify the composite function: (f ∘ g)(x) = f(g(x)) = 1/(√(x + 2) - 3) Step 2: Find domain of g(x): √(x + 2) requires x + 2 ≥ 0, so x ≥ -2 Step 3: Find domain of f(g(x)): f(g(x)) = 1/(√(x + 2) - 3) requires √(x + 2) - 3 ≠ 0 Step 4: Solve √(x + 2) - 3 ≠ 0: √(x + 2) ≠ 3, x + 2 ≠ 9, x ≠ 7 Step 5: Combine restrictions: x ≥ -2 AND x ≠ 7 Step 6: Write in interval notation: [-2, 7) ∪ (7, ∞) The domain is [-2, 7) ∪ (7, ∞)

6. 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.

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 minutes. 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: (3, ∞) Solution: For composite functions, the domain consists of all inputs where the inner function produces outputs that are valid inputs for the outer function.
   Full step-by-step solution

   For composite functions, the domain consists of all inputs where the inner function produces outputs that are valid inputs for the outer function. When dealing with square roots, the expression inside must be non-negative. For rational functions, the denominator cannot be zero. These restrictions combine to determine the final domain of the composition.