Function Operations

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Function Operations · Grade 12 · Worksheet 1

1. Aroha is sketching the graph of the rational function formed by combining two polynomial functions: f(x) = x^3 - 9x and g(x) = x^2 - 3x. The graph of h(x) = (f/g)(x) has vertical asymptotes and a hole at certain x-values. Determine the coordinates (x, y) of the hole in the graph of h(x). Answer: (3, 6) Solution: Factor f(x) = x^3 - 9x = x(x^2 - 9) = x(x - 3)(x + 3). Factor g(x) = x^2 - 3x = x(x - 3). Write h(x) = [x(x - 3)(x + 3)] / [x(x - 3)].
   Full step-by-step solution

   Step 1: Factor f(x) = x^3 - 9x = x(x^2 - 9) = x(x - 3)(x + 3). Step 2: Factor g(x) = x^2 - 3x = x(x - 3). Step 3: Write h(x) = [x(x - 3)(x + 3)] / [x(x - 3)]. Step 4: The common factors x and (x - 3) cancel, leaving h(x) = (x + 3) for x ≠ 0 and x ≠ 3. Step 5: The canceled factor (x - 3) gives a hole at x = 3. (The canceled factor x gives a hole at x = 0, but the problem asks for the hole from the common factor; we identify the one from the factor (x-3)). Step 6: Find the y-coordinate by substituting x = 3 into the simplified function: h(3) = 3 + 3 = 6. Step 7: The hole is at (3, 6).

2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is C(t) = 5t²e^(-0.2t) mg/L, where t is time in hours. To determine the maximum concentration and when it occurs, the research team needs to find the critical points of this function. Find the derivative C'(t) and determine at what time the drug concentration reaches its maximum value. Answer: 10 hours Solution: Identify the function and the rule for differentiation. C(t) = 5t² e^(-0.2t) This is a product of two functions: u(t) = 5t² and v(t) = e^(-0.2t). The product rule says: (u v)' = u' v + u v' Differentiate u(t) and v(t).
   Full step-by-step solution

   Let's find the derivative of C(t) = 5t² e^(-0.2t) and determine when the concentration is maximum. Step 1: Identify the function and the rule for differentiation. C(t) = 5t² e^(-0.2t) This is a product of two functions: u(t) = 5t² and v(t) = e^(-0.2t). The product rule says: (u v)' = u' v + u v' Step 2: Differentiate u(t) and v(t). u(t) = 5t² u'(t) = 10t v(t) = e^(-0.2t) The derivative of e^(k t) is k e^(k t). Here k = -0.2. So v'(t) = -0.2 e^(-0.2t) Step 3: Apply the product rule. C'(t) = u' v + u v' C'(t) = (10t) e^(-0.2t) + (5t²) (-0.2 e^(-0.2t)) Step 4: Simplify the expression. C'(t) = 10t e^(-0.2t) - 1.0 t² e^(-0.2t) Factor out e^(-0.2t) and t: C'(t) = e^(-0.2t) [10t - t²] Or: C'(t) = t e^(-0.2t) (10 - t) Step 5: Find critical points by setting C'(t) = 0. C'(t) = t e^(-0.2t) (10 - t) = 0 Since e^(-0.2t) is never zero for any finite t, we solve: t (10 - t) = 0 So t = 0 or t = 10. Step 6: Determine which critical point gives the maximum. At t = 0, the concentration C(0) = 0. At t = 10, we expect a maximum because for t between 0 and 10, C'(t) is positive (check: t>0 and (10-t)>0 so C'(t)>0), meaning C(t) is increasing. For t > 10, (10-t) is negative, so C'(t) < 0, meaning C(t) is decreasing. Thus, t = 10 is a local maximum. Step 7: Conclusion. The drug concentration reaches its maximum value at t = 10 hours. ANSWER: 10 hours

3. Aroha is a marine ecologist studying two species of algae in a coastal ecosystem. The biomass of Species X after t days is modeled by f(t) = 12t^2 + 9t + 15 kilograms, and the biomass of Species Y is modeled by g(t) = 9t + 10 kilograms. Aroha wants to understand how the total biomass of both species combined changes over time, and also the ratio of the biomass of Species X to Species Y. Determine the combined biomass function h(t) = (f + g)(t) and the ratio function r(t) = (f / g)(t). Then find the total combined biomass when t = 10 days. Answer: h(10) = 1335 Solution: Write the given functions: f(t) = 12t^2 + 9t + 15, g(t) = 9t + 10. Find h(t) = (f + g)(t) by adding the expressions: h(t) = (12t^2 + 9t + 15) + (9t + 10). Combine like terms: 12t^2 remains alone.
   Full step-by-step solution

   Step 1: Write the given functions: f(t) = 12t^2 + 9t + 15, g(t) = 9t + 10. Step 2: Find h(t) = (f + g)(t) by adding the expressions: h(t) = (12t^2 + 9t + 15) + (9t + 10). Step 3: Combine like terms: 12t^2 remains alone. For t terms: 9t + 9t = 18t. Constant terms: 15 + 10 = 25. Step 4: So h(t) = 12t^2 + 18t + 25. Step 5: Find r(t) = (f / g)(t) = (12t^2 + 9t + 15) / (9t + 10). This is already in simplest form since the numerator does not factor with a factor of (9t + 10). The domain excludes t where g(t) = 0, i.e., 9t + 10 = 0 => t = -10/9, which is not relevant for non-negative time. Step 6: Now find h(10): substitute t = 10 into h(t): h(10) = 12(10)^2 + 18(10) + 25. Step 7: Calculate: 12(100) = 1200; 18(10) = 180; then 1200 + 180 + 25 = 1405. The answer is h(10) = 1405.

4. f(x) = 3x² - 5x + 10, g(x) = 2x² + 5x - 15. Find (f - g)(x) = ? Answer: x² - 10x + 25 Solution: Write the expression for (f - g)(x) = f(x) - g(x) Substitute the functions: (3x² - 5x + 10) - (2x² + 5x - 15) Distribute the negative sign: 3x² - 5x + 10 - 2x² - 5x + 15 - x² terms: 3x² - 2x² = x² - x terms: -5x - 5x = -10x - constant terms: 10 + 15 = 25 Write the final expression: x² - 10x + 25…
   Full step-by-step solution

   Step 1: Write the expression for (f - g)(x) = f(x) - g(x) Step 2: Substitute the functions: (3x² - 5x + 10) - (2x² + 5x - 15) Step 3: Distribute the negative sign: 3x² - 5x + 10 - 2x² - 5x + 15 Step 4: Combine like terms: - x² terms: 3x² - 2x² = x² - x terms: -5x - 5x = -10x - constant terms: 10 + 15 = 25 Step 5: Write the final expression: x² - 10x + 25 The answer is x² - 10x + 25.

5. Isabella is an environmental engineer modeling the water flow rate through two different filtration systems at a water treatment plant. The flow rate through the first filter is modeled by f(x) = 7x^3 - 2x^2 + 12x - 7, where x represents the pressure setting. The flow rate through the second filter is modeled by g(x) = -2x^3 + 17x^2 + 2x - 2. To determine the combined flow rate when both filters are operating simultaneously, Isabella needs to find the function h(x) = f(x) + g(x). Determine the simplified function h(x). Answer: 5x^3 + 15x^2 + 14x - 9 Solution: Write the two functions: f(x) = 7x^3 - 2x^2 + 12x - 7 and g(x) = -2x^3 + 17x^2 + 2x - 2. Add the functions: h(x) = f(x) + g(x) = (7x^3 - 2x^2 + 12x - 7) + (-2x^3 + 17x^2 + 2x - 2).
   Full step-by-step solution

   Step 1: Write the two functions: f(x) = 7x^3 - 2x^2 + 12x - 7 and g(x) = -2x^3 + 17x^2 + 2x - 2. Step 2: Add the functions: h(x) = f(x) + g(x) = (7x^3 - 2x^2 + 12x - 7) + (-2x^3 + 17x^2 + 2x - 2). Step 3: Combine like terms (terms with the same power of x): - x^3 terms: 7x^3 + (-2x^3) = 5x^3 - x^2 terms: -2x^2 + 17x^2 = 15x^2 - x terms: 12x + 2x = 14x - Constant terms: -7 + (-2) = -9 Step 4: Write the simplified result: h(x) = 5x^3 + 15x^2 + 14x - 9. The answer is 5x^3 + 15x^2 + 14x - 9.

6. 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

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