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Function Operations

Grade 12 · Algebra · Worksheet 1

  1. Given the functions f(x) = 2x^3 - 5x and g(x) = 4e^(3x), find the exact value of the derivative h'(0) where h(x) = f(x) * g(x). Answer: ______________
  2. f(x) = 3x³ - 5x + 7, g(x) = 2x³ + x - 9. Find (f - g)(x) = ? Answer: ______________
  3. A marine biologist is studying the population growth of an endangered coral species. The population function is P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)), where t is time in years since monitoring began. The conservation effectiveness is modeled by E(p) = ln(p/500), where p is the population size. What is the rate of change of conservation effectiveness with respect to time when t = 20 years? Answer: ______________
  4. f(x) = 5x³ - 9x² + 12, g(x) = 2x³ + 7x - 4. Find (f - g)(x) = ? Answer: ______________
  5. f(x) = 2x² - 4x + 8, g(x) = 6x² + 2x - 12. Find (f + g)(x) = ? Answer: ______________
  6. f(x) = 4x³ - 5x² + 10, g(x) = 2x³ + 3x - 15. Find (f + g)(x) Answer: ______________
  7. f(x) = 2x³ - 5x + 10, g(x) = 3x² + 4x - 15. Find (f + g)(x) Answer: ______________
  8. f(x) = 2x³ - 5x² + 11, g(x) = 4x² + 7x - 13. Find (f + g)(x) = ? Answer: ______________
  9. f(x) = 3x⁴ - 8x² + 12, g(x) = 2x⁴ + 5x² - 9. Find (f - g)(x) = ? Answer: ______________
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Answer Key & Explanations

Function Operations · Grade 12 · Worksheet 1

  1. Given the functions f(x) = 2x^3 - 5x and g(x) = 4e^(3x), find the exact value of the derivative h'(0) where h(x) = f(x) * g(x). Answer: -20 Solution: Identify h(x) = f(x) * g(x) = (2x^3 - 5x) * (4e^(3x)) Apply the product rule: h'(x) = f'(x)g(x) + f(x)g'(x) Find f'(x) = derivative of (2x^3 - 5x) = 6x^2 - 5 Find g'(x) = derivative of (4e^(3x)) = 12e^(3x) Substitute into product rule: h'(x) = (6x^2 - 5)(4e^(3x)) + (2x^3 - 5x)(12e^(3x)) Evaluate…
    Full step-by-step solution

    Step 1: Identify h(x) = f(x) * g(x) = (2x^3 - 5x) * (4e^(3x)) Step 2: Apply the product rule: h'(x) = f'(x)g(x) + f(x)g'(x) Step 3: Find f'(x) = derivative of (2x^3 - 5x) = 6x^2 - 5 Step 4: Find g'(x) = derivative of (4e^(3x)) = 12e^(3x) Step 5: Substitute into product rule: h'(x) = (6x^2 - 5)(4e^(3x)) + (2x^3 - 5x)(12e^(3x)) Step 6: Evaluate at x = 0: h'(0) = (6(0)^2 - 5)(4e^(0)) + (2(0)^3 - 5(0))(12e^(0)) Step 7: Simplify: h'(0) = (-5)(4*1) + (0)(12*1) Step 8: Calculate: h'(0) = (-5)(4) + 0 = -20 The answer is -20.

  2. f(x) = 3x³ - 5x + 7, g(x) = 2x³ + x - 9. Find (f - g)(x) = ? Answer: x³ - 6x + 16 Solution: Write the expression for (f - g)(x): (3x³ - 5x + 7) - (2x³ + x - 9) Distribute the negative sign: 3x³ - 5x + 7 - 2x³ - x + 9 Combine like terms for x³: 3x³ - 2x³ = x³ Combine like terms for x: -5x - x = -6x Combine constant terms: 7 + 9 = 16 Write the final result: x³ - 6x + 16 The answer is x³…
    Full step-by-step solution

    Step 1: Write the expression for (f - g)(x): (3x³ - 5x + 7) - (2x³ + x - 9) Step 2: Distribute the negative sign: 3x³ - 5x + 7 - 2x³ - x + 9 Step 3: Combine like terms for x³: 3x³ - 2x³ = x³ Step 4: Combine like terms for x: -5x - x = -6x Step 5: Combine constant terms: 7 + 9 = 16 Step 6: Write the final result: x³ - 6x + 16 The answer is x³ - 6x + 16.

  3. A marine biologist is studying the population growth of an endangered coral species. The population function is P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)), where t is time in years since monitoring began. The conservation effectiveness is modeled by E(p) = ln(p/500), where p is the population size. What is the rate of change of conservation effectiveness with respect to time when t = 20 years? Answer: 0.015 Solution: We need to find dE/dt when t = 20. Since E is a function of p and p is a function of t, we use the chain rule: dE/dt = (dE/dp) × (dp/dt). First, find dE/dp.
    Full step-by-step solution

    Step 1: We need to find dE/dt when t = 20. Since E is a function of p and p is a function of t, we use the chain rule: dE/dt = (dE/dp) × (dp/dt). Step 2: First, find dE/dp. E(p) = ln(p/500) = ln(p) - ln(500). So dE/dp = 1/p. Step 3: Now find dp/dt. P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)). Using the quotient rule: dp/dt = [2000(0.03)e^(0.03t)(1 + 0.5e^(0.03t)) - 2000e^(0.03t)(0.5)(0.03)e^(0.03t)] / (1 + 0.5e^(0.03t))^2. Step 4: Simplify: dp/dt = [60e^(0.03t)(1 + 0.5e^(0.03t)) - 30e^(0.06t)] / (1 + 0.5e^(0.03t))^2. Step 5: When t = 20, first calculate e^(0.03×20) = e^(0.6) ≈ 1.8221. Then p(20) = 2000×1.8221 / (1 + 0.5×1.8221) = 3644.2 / (1 + 0.91105) = 3644.2 / 1.91105 ≈ 1907.5. Step 6: Now calculate dp/dt at t = 20: dp/dt = [60×1.8221×(1 + 0.91105) - 30×(1.8221)^2] / (1.91105)^2 = [109.326×(1.91105) - 30×3.320] / 3.6521 = [208.9 - 99.6] / 3.6521 = 109.3 / 3.6521 ≈ 29.93. Step 7: Now dE/dt = (dE/dp) × (dp/dt) = (1/p) × (dp/dt) = (1/1907.5) × 29.93 ≈ 0.000524 × 29.93 ≈ 0.0157. Step 8: Rounded to three decimal places, the rate of change is 0.015. The answer is 0.015.

  4. f(x) = 5x³ - 9x² + 12, g(x) = 2x³ + 7x - 4. Find (f - g)(x) = ? Answer: 3x³ - 9x² - 7x + 16 Solution: Write the expression for (f - g)(x): (5x³ - 9x² + 12) - (2x³ + 7x - 4) Distribute the negative sign: 5x³ - 9x² + 12 - 2x³ - 7x + 4 Combine like terms for x³: 5x³ - 2x³ = 3x³ The -9x² term has no like term, so it remains: -9x² Combine the x terms: -7x (no other x terms) Combine constant terms: 12…
    Full step-by-step solution

    Step 1: Write the expression for (f - g)(x): (5x³ - 9x² + 12) - (2x³ + 7x - 4) Step 2: Distribute the negative sign: 5x³ - 9x² + 12 - 2x³ - 7x + 4 Step 3: Combine like terms for x³: 5x³ - 2x³ = 3x³ Step 4: The -9x² term has no like term, so it remains: -9x² Step 5: Combine the x terms: -7x (no other x terms) Step 6: Combine constant terms: 12 + 4 = 16 Step 7: Write the final expression: 3x³ - 9x² - 7x + 16

  5. f(x) = 2x² - 4x + 8, g(x) = 6x² + 2x - 12. Find (f + g)(x) = ? Answer: 8x² - 2x - 4 Solution: Write the addition of the two functions: (f + g)(x) = (2x² - 4x + 8) + (6x² + 2x - 12) Combine x² terms: 2x² + 6x² = 8x² Combine x terms: -4x + 2x = -2x Combine constant terms: 8 + (-12) = 8 - 12 = -4 Write the final expression: 8x² - 2x - 4 The answer is 8x² - 2x - 4.
    Full step-by-step solution

    Step 1: Write the addition of the two functions: (f + g)(x) = (2x² - 4x + 8) + (6x² + 2x - 12) Step 2: Combine x² terms: 2x² + 6x² = 8x² Step 3: Combine x terms: -4x + 2x = -2x Step 4: Combine constant terms: 8 + (-12) = 8 - 12 = -4 Step 5: Write the final expression: 8x² - 2x - 4 The answer is 8x² - 2x - 4.

  6. f(x) = 4x³ - 5x² + 10, g(x) = 2x³ + 3x - 15. Find (f + g)(x) Answer: 6x³ - 5x² + 3x - 5 Solution: Write the sum of the functions: (f + g)(x) = (4x³ - 5x² + 10) + (2x³ + 3x - 15) Group like terms: (4x³ + 2x³) + (-5x²) + (3x) + (10 - 15) Add coefficients of x³ terms: 4 + 2 = 6, so 6x³ The -5x² term has no like term in g(x), so it remains -5x² The 3x term has no like term in f(x), so it remains…
    Full step-by-step solution

    Step 1: Write the sum of the functions: (f + g)(x) = (4x³ - 5x² + 10) + (2x³ + 3x - 15) Step 2: Group like terms: (4x³ + 2x³) + (-5x²) + (3x) + (10 - 15) Step 3: Add coefficients of x³ terms: 4 + 2 = 6, so 6x³ Step 4: The -5x² term has no like term in g(x), so it remains -5x² Step 5: The 3x term has no like term in f(x), so it remains 3x Step 6: Add constant terms: 10 + (-15) = -5 Step 7: Combine all terms: 6x³ - 5x² + 3x - 5 The answer is 6x³ - 5x² + 3x - 5.

  7. f(x) = 2x³ - 5x + 10, g(x) = 3x² + 4x - 15. Find (f + g)(x) Answer: 2x³ + 3x² - x - 5 Solution: Write the addition: (f + g)(x) = (2x³ - 5x + 10) + (3x² + 4x - 15) Group like terms: 2x³ + 3x² + (-5x + 4x) + (10 - 15) Combine x³ terms: 2x³ Combine x² terms: + 3x² Combine x terms: -5x + 4x = -x Combine constant terms: 10 - 15 = -5 Final result: 2x³ + 3x² - x - 5
    Full step-by-step solution

    Step 1: Write the addition: (f + g)(x) = (2x³ - 5x + 10) + (3x² + 4x - 15) Step 2: Group like terms: 2x³ + 3x² + (-5x + 4x) + (10 - 15) Step 3: Combine x³ terms: 2x³ Step 4: Combine x² terms: + 3x² Step 5: Combine x terms: -5x + 4x = -x Step 6: Combine constant terms: 10 - 15 = -5 Step 7: Final result: 2x³ + 3x² - x - 5

  8. f(x) = 2x³ - 5x² + 11, g(x) = 4x² + 7x - 13. Find (f + g)(x) = ? Answer: 2x³ - x² + 7x - 2 Solution: Write out the addition: (f + g)(x) = (2x³ - 5x² + 11) + (4x² + 7x - 13) Group like terms: x³ terms: 2x³, x² terms: -5x² + 4x², x terms: 7x, constant terms: 11 + (-13) Add coefficients: 2x³ + (-5x² + 4x²) + 7x + (11 - 13) Simplify: 2x³ + (-1x²) + 7x + (-2) Final answer: 2x³ - x² + 7x - 2
    Full step-by-step solution

    Step 1: Write out the addition: (f + g)(x) = (2x³ - 5x² + 11) + (4x² + 7x - 13) Step 2: Group like terms: x³ terms: 2x³, x² terms: -5x² + 4x², x terms: 7x, constant terms: 11 + (-13) Step 3: Add coefficients: 2x³ + (-5x² + 4x²) + 7x + (11 - 13) Step 4: Simplify: 2x³ + (-1x²) + 7x + (-2) Step 5: Final answer: 2x³ - x² + 7x - 2

  9. f(x) = 3x⁴ - 8x² + 12, g(x) = 2x⁴ + 5x² - 9. Find (f - g)(x) = ? Answer: x⁴ - 13x² + 21 Solution: Write the expression for (f - g)(x): (3x⁴ - 8x² + 12) - (2x⁴ + 5x² - 9) Distribute the negative sign: 3x⁴ - 8x² + 12 - 2x⁴ - 5x² + 9 Combine like terms for x⁴ terms: 3x⁴ - 2x⁴ = x⁴ Combine like terms for x² terms: -8x² - 5x² = -13x² Combine constant terms: 12 + 9 = 21 Write the final result: x⁴…
    Full step-by-step solution

    Step 1: Write the expression for (f - g)(x): (3x⁴ - 8x² + 12) - (2x⁴ + 5x² - 9) Step 2: Distribute the negative sign: 3x⁴ - 8x² + 12 - 2x⁴ - 5x² + 9 Step 3: Combine like terms for x⁴ terms: 3x⁴ - 2x⁴ = x⁴ Step 4: Combine like terms for x² terms: -8x² - 5x² = -13x² Step 5: Combine constant terms: 12 + 9 = 21 Step 6: Write the final result: x⁴ - 13x² + 21