Rational Functions

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Rational Functions · Grade 12 · Worksheet 2

1. An environmental engineer is modeling the population growth of an endangered species in a protected habitat. The population function is given by P(t) = (2t² + 5t - 3)/(t² - 9), where P(t) represents the population in hundreds and t represents years since the conservation program began. The conservation team needs to understand the long-term carrying capacity of the habitat and identify any critical time points where the population model becomes undefined. Determine all vertical and horizontal asymptotes of this population function. Answer: x = -3, x = 3, y = 2 Solution: Horizontal asymptotes show the long-term behavior or limiting value of the system.
   Full step-by-step solution

   Rational functions often model real-world scenarios like population growth, chemical concentrations, or economic trends. Vertical asymptotes represent values where the model becomes undefined, which might correspond to impossible situations in the real world. Horizontal asymptotes show the long-term behavior or limiting value of the system. When finding asymptotes, we analyze where the denominator equals zero for vertical asymptotes, and compare the degrees of the numerator and denominator polynomials for horizontal asymptotes.

2. 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 is concentration in mg/L and t is time in hours. The medical team needs to understand the long-term behavior of the drug concentration and identify any time points where the concentration becomes undefined. Determine the horizontal asymptote of this function and identify the vertical asymptotes. Answer: Horizontal asymptote: y = 3; Vertical asymptotes: t = 2 and t = -2 Solution: C(t) = (3t² + 5t) / (t² - 4) Vertical asymptotes occur where the denominator is zero (and numerator is not also zero at the same point, which would indicate a possible hole).
   Full step-by-step solution

   Let's go step-by-step. We have the function: C(t) = (3t² + 5t) / (t² - 4) --- **Step 1: Identify vertical asymptotes** Vertical asymptotes occur where the denominator is zero (and numerator is not also zero at the same point, which would indicate a possible hole). Denominator: t² - 4 = 0 t² = 4 t = 2 or t = -2 Check numerator at these points: At t = 2: 3(2)² + 5(2) = 12 + 10 = 22 ≠ 0 At t = -2: 3(4) + 5(-2) = 12 - 10 = 2 ≠ 0 So both make the denominator 0 but not numerator → vertical asymptotes. Vertical asymptotes: t = 2 and t = -2 --- **Step 2: Find horizontal asymptote** For rational functions, compare degrees of numerator and denominator. Numerator degree = 2 (leading term 3t²) Denominator degree = 2 (leading term t²) When degrees are equal, horizontal asymptote is the ratio of leading coefficients. Leading coefficient of numerator = 3 Leading coefficient of denominator = 1 Horizontal asymptote: y = 3/1 = 3 --- **Step 3: Conclusion** Horizontal asymptote: y = 3 Vertical asymptotes: t = 2 and t = -2 --- Final answer: Horizontal asymptote: y = 3; Vertical asymptotes: t = 2 and t = -2

3. An environmental engineer is modeling the rate of pollutant absorption in a wetland using the function R(x) = (2x² - 5x + 3)/(x² - 9), where R(x) represents the absorption rate in grams per square meter per day and x represents the distance from the pollution source in kilometers. The engineering team needs to understand the long-term absorption rate as distance increases and identify any distances where the absorption model becomes undefined. Determine all vertical and horizontal asymptotes of this absorption rate function. Answer: Horizontal asymptote: y = 2; Vertical asymptotes: x = 3 and x = -3 Solution: Find vertical asymptotes by setting the denominator equal to zero: x² - 9 = 0 Factor the denominator: (x - 3)(x + 3) = 0 Solve for x: x = 3 and x = -3 These are vertical asymptotes since they don't cancel with numerator factors Find horizontal asymptote by comparing degrees of numerator and…
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero: x² - 9 = 0 Step 2: Factor the denominator: (x - 3)(x + 3) = 0 Step 3: Solve for x: x = 3 and x = -3 Step 4: These are vertical asymptotes since they don't cancel with numerator factors Step 5: Find horizontal asymptote by comparing degrees of numerator and denominator Step 6: Both numerator and denominator are degree 2 polynomials Step 7: Horizontal asymptote is the ratio of leading coefficients: 2/1 = 2 Step 8: Therefore, the horizontal asymptote is y = 2 Final answer: Horizontal asymptote: y = 2; Vertical asymptotes: x = 3 and x = -3

4. lim_{x→∞} (3x² - 2x + 5)/(x² + 4x - 1) = ? Answer: 3 Solution: limit as x → ∞ of (3x² - 2x + 5) / (x² + 4x - 1) Identify the highest power of x in the denominator The highest power of x in both numerator and denominator is x².
   Full step-by-step solution

   Let's solve step by step. We want: limit as x → ∞ of (3x² - 2x + 5) / (x² + 4x - 1) --- **Step 1: Identify the highest power of x in the denominator** The highest power of x in both numerator and denominator is x². --- **Step 2: Divide every term in numerator and denominator by x²** Numerator: (3x² - 2x + 5) / x² = 3x²/x² - 2x/x² + 5/x² = 3 - 2/x + 5/x² Denominator: (x² + 4x - 1) / x² = x²/x² + 4x/x² - 1/x² = 1 + 4/x - 1/x² So the expression becomes: [3 - 2/x + 5/x²] / [1 + 4/x - 1/x²] --- **Step 3: Take the limit as x → ∞** As x → ∞: - 2/x → 0 - 5/x² → 0 - 4/x → 0 - 1/x² → 0 So the expression approaches: (3 - 0 + 0) / (1 + 0 - 0) = 3/1 = 3 --- **Step 4: Conclusion** The limit is 3.

5. An environmental scientist is modeling the population growth of an endangered species using the function P(t) = (2t² + 5t - 3)/(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 habitat and identify any 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 vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 9 = 0 Factor: (t - 3)(t + 3) = 0 So t = 3 and t = -3 are vertical asymptotes.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 9 = 0 Factor: (t - 3)(t + 3) = 0 So t = 3 and t = -3 are vertical asymptotes. Step 2: Find horizontal asymptote by comparing degrees of numerator and denominator. Both numerator and denominator are degree 2 polynomials. Horizontal asymptote = ratio of leading coefficients. Leading coefficient of numerator: 2 Leading coefficient of denominator: 1 Horizontal asymptote: y = 2/1 = 2 Step 3: Final answer: Horizontal asymptote is y = 2; Vertical asymptotes are t = 3 and t = -3.

6. Consider the rational function f(x) = (2x^4 - 3x^3 + 5x - 1)/(4x^4 + x^2 - 7). Determine the horizontal asymptote of this function. If the asymptote is a horizontal line, provide the y-value of that line as your answer. Answer: 0.5 Solution: Identify the degrees of the numerator and denominator. Both the numerator and denominator are degree 4 polynomials.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. Both the numerator and denominator are degree 4 polynomials. Step 2: When the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 2 (from 2x^4). Step 4: The leading coefficient of the denominator is 4 (from 4x^4). Step 5: Calculate the ratio: 2/4 = 1/2 = 0.5 Step 6: Therefore, the horizontal asymptote is y = 0.5 The answer is 0.5.