Rational Functions

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Answer Key & Explanations

Rational Functions · Grade 12 · Worksheet 1

1. lim(x→∞) (4x³ - 2x² + 7)/(3x³ + 5x - 1) = ? Answer: 4/3 Solution: Identify the degrees of numerator and denominator. Both are degree 3. Since degrees are equal, the limit equals the ratio of leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 3. Step 2: Since degrees are equal, the limit equals the ratio of leading coefficients. Step 3: Leading coefficient of numerator is 4, leading coefficient of denominator is 3. Step 4: The limit is 4/3. The answer is 4/3.

2. lim(x→∞) (5x³ - 2x² + 3x - 7)/(4x³ + x² - 5x + 2) = ? Answer: 5/4 Solution: Identify the degrees of numerator and denominator. Both are degree 3. For rational functions where degrees are equal, the limit at infinity equals the ratio of leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 3. Step 2: For rational functions where degrees are equal, the limit at infinity equals the ratio of leading coefficients. Step 3: Leading coefficient of numerator is 5, leading coefficient of denominator is 4. Step 4: The limit is 5/4. The answer is 5/4.

3. lim(x→∞) (5x³ - 2x² + 3x - 1)/(4x³ + x² - 7) = ? Answer: 5/4 Solution: Identify the degrees of numerator and denominator. Both are degree 3. For rational functions where degrees are equal, the limit at infinity equals the ratio of the leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 3. Step 2: For rational functions where degrees are equal, the limit at infinity equals the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 5. Step 4: The leading coefficient of the denominator is 4. Step 5: The limit equals 5/4. The answer is 5/4.

4. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time using the function C(t) = (3t² + 5t) / (t² - 4), where C(t) is the concentration in milligrams per liter and t is time in hours. The research team needs to determine the long-term concentration level and identify any time points where the concentration becomes undefined. What is the horizontal asymptote of this function, and at what time values does the concentration become undefined? Answer: Horizontal asymptote: y = 3; Undefined at t = 2 and t = -2 Solution: C(t) = (3t² + 5t) / (t² - 4) A rational function is undefined where the denominator is zero. Set denominator = 0: t² - 4 = 0 t² = 4 t = 2 or t = -2 So the concentration is undefined at t = 2 hours and t = -2 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify where the function is undefined** The function is C(t) = (3t² + 5t) / (t² - 4) A rational function is undefined where the denominator is zero. Set denominator = 0: t² - 4 = 0 t² = 4 t = 2 or t = -2 So the concentration is undefined at t = 2 hours and t = -2 hours. (Since time can't be negative in reality, t = -2 is mathematically an undefined point but not physically meaningful.) --- **Step 2: Find the horizontal asymptote** For a rational function (polynomial / polynomial): If degree of numerator = degree of denominator (both degree 2 here), horizontal asymptote = ratio of leading coefficients. Numerator leading term: 3t² Denominator leading term: t² Ratio = 3/1 = 3 So horizontal asymptote: y = 3 This means as t → ∞ or t → -∞, C(t) → 3 mg/L. --- **Step 3: Final answer** Horizontal asymptote: y = 3 Undefined at t = 2 and t = -2 --- **Final Answer:** Horizontal asymptote: y = 3; Undefined at t = 2 and t = -2

5. Hana is a marine biologist studying the feeding rate of a filter-feeding organism in a tidal estuary. She models the rate at which the organism filters water using the rational function F(t) = (4t² + 8t - 12)/(t² - 16), where F(t) represents the filtration rate in liters per hour and t represents the time in hours after low tide. Hana needs to understand the long-term filtration rate as time increases and identify any times when the model becomes undefined due to tidal changes. Determine all vertical asymptotes and the horizontal asymptote of this filtration rate function, and describe the behavior of F(t) near each vertical asymptote. Answer: Vertical asymptotes at t = -4 and t = 4; horizontal asymptote at y = 4. As t approaches -4 from the left, F(t) approaches +infinity; from the right, -infinity. As t approaches 4 from the left, F(t) approaches -infinity; from the right, +infinity. Solution: Factor the numerator and denominator. Numerator: 4t² + 8t - 12 = 4(t² + 2t - 3) = 4(t + 3)(t - 1). Denominator: t² - 16 = (t - 4)(t + 4).
   Full step-by-step solution

   Step 1: Factor the numerator and denominator. Numerator: 4t² + 8t - 12 = 4(t² + 2t - 3) = 4(t + 3)(t - 1). Denominator: t² - 16 = (t - 4)(t + 4). So F(t) = 4(t + 3)(t - 1)/((t - 4)(t + 4)). Step 2: Find vertical asymptotes. Set denominator equal to zero: (t - 4)(t + 4) = 0, so t = 4 and t = -4. Check that numerator is not zero at these points. At t = 4: 4(4 + 3)(4 - 1) = 4(7)(3) = 84 ≠ 0. At t = -4: 4(-4 + 3)(-4 - 1) = 4(-1)(-5) = 20 ≠ 0. So both are vertical asymptotes. Step 3: Determine behavior near t = 4. As t approaches 4 from the left (e.g., t = 3.9), denominator is (3.9 - 4)(3.9 + 4) = (-0.1)(7.9) which is negative, numerator is positive, so F(t) approaches -infinity. As t approaches 4 from the right (e.g., t = 4.1), denominator is (0.1)(8.1) positive, numerator positive, so F(t) approaches +infinity. Step 4: Determine behavior near t = -4. As t approaches -4 from the left (e.g., t = -4.1), denominator is (-4.1 - 4)(-4.1 + 4) = (-8.1)(-0.1) = positive, numerator at t = -4.1 is 4(-1.1)(-5.1) = positive, so F(t) approaches +infinity. As t approaches -4 from the right (e.g., t = -3.9), denominator is (-3.9 - 4)(-3.9 + 4) = (-7.9)(0.1) = negative, numerator positive, so F(t) approaches -infinity. Step 5: Find horizontal asymptote. Both numerator and denominator are degree 2. The leading coefficient ratio is 4/1 = 4. So horizontal asymptote is y = 4. Final answer: Vertical asymptotes at t = -4 and t = 4; horizontal asymptote at y = 4. Near t = -4: as t → -4⁻, F(t) → +∞; as t → -4⁺, F(t) → -∞. Near t = 4: as t → 4⁻, F(t) → -∞; as t → 4⁺, F(t) → +∞.

6. An environmental engineer is modeling the population growth of an endangered species in a wildlife reserve using the function P(t) = (2t² - 8t + 6)/(t² - 9), where P(t) represents the population in hundreds and t represents time in years since conservation efforts began. The conservation team needs to understand the long-term carrying capacity of the reserve and identify any critical time points where the population model becomes undefined. Determine all vertical and horizontal asymptotes of this population function. Answer: Horizontal asymptote: y = 2; Vertical asymptotes: t = 3 and t = -3 Solution: Find horizontal asymptote by comparing degrees of numerator and denominator Both numerator and denominator have degree 2, so horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator) = 2/1 = 2 Find vertical asymptotes by setting denominator equal to zero…
   Full step-by-step solution

   Step 1: Find horizontal asymptote by comparing degrees of numerator and denominator Both numerator and denominator have degree 2, so horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator) = 2/1 = 2 Step 2: Find vertical asymptotes by setting denominator equal to zero t² - 9 = 0 (t - 3)(t + 3) = 0 t = 3 or t = -3 Step 3: Verify these are vertical asymptotes (not holes) Check if numerator is also zero at these points: At t = 3: 2(3)² - 8(3) + 6 = 18 - 24 + 6 = 0 At t = -3: 2(-3)² - 8(-3) + 6 = 18 + 24 + 6 = 48 Since numerator is not zero at both points, both are vertical asymptotes Final answer: Horizontal asymptote: y = 2; Vertical asymptotes: t = 3 and t = -3