Rational Functions

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Rational Functions · Grade 12 · Worksheet 3

1. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time using the rational function C(t) = (3t² + 12t) / (t² + 4t + 3), where C(t) represents the concentration in milligrams per liter and t represents time in hours after administration. The researchers need to determine the long-term concentration level that the medication approaches and identify any times when the concentration becomes undefined due to vertical asymptotes. What is the horizontal asymptote representing the long-term concentration, and at what time values does the concentration become undefined? Answer: Horizontal asymptote: y = 3; Vertical asymptotes: t = -3 and t = -1 Solution: C(t) = (3t² + 12t) / (t² + 4t + 3) Find the horizontal asymptote (long-term concentration) For a rational function (polynomial / polynomial), - If degree of numerator = degree of denominator, horizontal asymptote = (leading coefficient of numerator) / (leading coefficient of denominator).
   Full step-by-step solution

   Let's go step-by-step. We are given: C(t) = (3t² + 12t) / (t² + 4t + 3) --- **Step 1: Find the horizontal asymptote (long-term concentration)** For a rational function (polynomial / polynomial), - If degree of numerator = degree of denominator, horizontal asymptote = (leading coefficient of numerator) / (leading coefficient of denominator). Here: Numerator: 3t² + 12t → degree 2, leading coefficient 3 Denominator: t² + 4t + 3 → degree 2, leading coefficient 1 So horizontal asymptote: y = 3/1 = 3. Thus, long-term concentration approaches 3 mg/L. --- **Step 2: Find when C(t) is undefined (vertical asymptotes)** C(t) is undefined where denominator = 0 and numerator ≠ 0 (unless factor cancels). Denominator: t² + 4t + 3 = 0 Factor: (t + 3)(t + 1) = 0 So t = -3 or t = -1. --- **Step 3: Check if these make numerator zero (possible hole instead of asymptote)** Numerator: 3t² + 12t = 3t(t + 4) At t = -3: 3(-3)(-3 + 4) = (-9)(1) = -9 ≠ 0 At t = -1: 3(-1)(-1 + 4) = (-3)(3) = -9 ≠ 0 So no cancellation; both are vertical asymptotes. --- **Step 4: Interpret time values** t is time in hours after administration. t = -3 and t = -1 are before administration (t=0), so in a real-world sense, the model may only be intended for t ≥ 0. But mathematically, vertical asymptotes are at t = -3 and t = -1. --- **Final Answer:** Horizontal asymptote: y = 3 Vertical asymptotes: t = -3 and t = -1

2. A civil engineer is designing a suspension bridge where the cable shape follows the rational function f(x) = (2x² - 8x + 6)/(x² - 9), where x represents the horizontal distance from the center of the bridge in meters and f(x) represents the cable height above the roadway. To ensure proper clearance and structural integrity, the engineer needs to determine the cable's long-term behavior and identify any positions where the height becomes undefined. Find all vertical and horizontal asymptotes of this bridge cable function. Answer: Vertical asymptotes: x = 3, x = -3; Horizontal asymptote: y = 2 Solution: Find vertical asymptotes by setting the denominator equal to zero x² - 9 = 0 x² = 9 x = 3 or x = -3 These are the vertical asymptotes.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero x² - 9 = 0 x² = 9 x = 3 or x = -3 These are the vertical asymptotes. Step 2: Find horizontal asymptote by comparing degrees of numerator and denominator Both numerator and denominator are degree 2 polynomials. When degrees are equal, horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator) Leading coefficient of numerator is 2 Leading coefficient of denominator is 1 Horizontal asymptote is y = 2/1 = 2 Step 3: Final answer Vertical asymptotes: x = 3, x = -3 Horizontal asymptote: y = 2

3. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration function is given by C(t) = (3t² + 5t - 2)/(t² - 4), where t represents hours after administration and C(t) is measured in milligrams per liter. The medical team needs to understand the long-term behavior of this medication to determine appropriate dosing intervals. What horizontal asymptote does this concentration function approach as time increases indefinitely? Answer: y = 3 Solution: To find the horizontal asymptote of the rational function C(t) = (3t² + 5t - 2)/(t² - 4) as t increases indefinitely, we compare the degrees of the numerator and denominator and examine the leading coefficients.
   Full step-by-step solution

   To find the horizontal asymptote of the rational function C(t) = (3t² + 5t - 2)/(t² - 4) as t increases indefinitely, we compare the degrees of the numerator and denominator and examine the leading coefficients. Step 1: Identify the degrees of the numerator and denominator. The numerator is 3t² + 5t - 2. The highest power of t is t², so the degree is 2. The denominator is t² - 4. The highest power of t is t², so the degree is also 2. Step 2: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator (the coefficient of t²) is 3. The leading coefficient of the denominator (the coefficient of t²) is 1. Step 3: Calculate the ratio of the leading coefficients. Horizontal asymptote = (leading coefficient of numerator) / (leading coefficient of denominator) = 3 / 1 = 3. Step 4: State the result. Therefore, as time t increases indefinitely, the concentration function C(t) approaches the horizontal asymptote y = 3. Final Answer: y = 3

4. A pharmaceutical company is modeling the concentration of a new medication in the bloodstream over time using the function C(t) = (5t + 10)/(t² + 4), where C(t) represents the concentration in milligrams per liter and t represents time in hours after administration. The researchers need to determine the long-term behavior of the drug concentration and identify any time points where the concentration might approach unrealistic values. Find all horizontal and vertical asymptotes of this rational function. Answer: Horizontal asymptote: y = 0; Vertical asymptotes: none Solution: A horizontal asymptote describes the behavior of C(t) as t → ∞ or t → -∞. For a rational function (polynomial)/(polynomial): - If the degree of numerator < degree of denominator, horizontal asymptote is y = 0.
   Full step-by-step solution

   Let's find the horizontal and vertical asymptotes of the function C(t) = (5t + 10)/(t² + 4). --- **Step 1: Identify horizontal asymptotes** A horizontal asymptote describes the behavior of C(t) as t → ∞ or t → -∞. For a rational function (polynomial)/(polynomial): - If the degree of numerator < degree of denominator, horizontal asymptote is y = 0. - If degrees are equal, horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). - If degree of numerator > degree of denominator, no horizontal asymptote (but maybe oblique asymptote). Here: Numerator: 5t + 10 → degree 1 Denominator: t² + 4 → degree 2 Since 1 < 2, the horizontal asymptote is y = 0. We can check by dividing numerator and denominator by t² (highest power in denominator): C(t) = (5t + 10)/(t² + 4) = (5/t + 10/t²)/(1 + 4/t²) As t → ∞: 5/t → 0, 10/t² → 0, 4/t² → 0, so C(t) → 0/1 = 0. Thus, horizontal asymptote: y = 0. --- **Step 2: Identify vertical asymptotes** Vertical asymptotes occur where the denominator is 0 and numerator is not 0 at the same t. Set denominator = 0: t² + 4 = 0 t² = -4 This has no real solutions (solutions are t = 2i and t = -2i, which are complex numbers). Since the denominator is never 0 for any real t, there are no vertical asymptotes. --- **Step 3: Conclusion** Horizontal asymptote: y = 0 Vertical asymptotes: none --- **Final answer:** Horizontal asymptote: y = 0; Vertical asymptotes: none

5. Consider the rational function f(x) = (3x^3 - 5x^2 + 2x - 7)/(2x^3 + 4x^2 - x + 1). As x approaches positive infinity, what value does f(x) approach? Provide your answer as a simplified fraction. Answer: 3/2 Solution: To determine the value that f(x) approaches as x approaches positive infinity, we analyze the rational function f(x) = (3x^3 - 5x^2 + 2x - 7) / (2x^3 + 4x^2 - x + 1).
   Full step-by-step solution

   To determine the value that f(x) approaches as x approaches positive infinity, we analyze the rational function f(x) = (3x^3 - 5x^2 + 2x - 7) / (2x^3 + 4x^2 - x + 1). Step 1: Identify the degrees of the numerator and denominator. - The numerator is 3x^3 - 5x^2 + 2x - 7. The highest power of x is x^3, so the degree is 3. - The denominator is 2x^3 + 4x^2 - x + 1. The highest power of x is x^3, so the degree is 3. Step 2: Since the degrees of the numerator and denominator are equal, the limit as x approaches infinity is the ratio of the leading coefficients. - The leading coefficient of the numerator is 3 (from 3x^3). - The leading coefficient of the denominator is 2 (from 2x^3). Step 3: Therefore, the limit is 3/2. Step 4: Explanation: As x becomes very large, the terms with the highest power (x^3) dominate the behavior of the function. The lower degree terms (-5x^2, +2x, -7 in the numerator and +4x^2, -x, +1 in the denominator) become negligible compared to the x^3 terms. So, for very large x, f(x) behaves approximately like (3x^3)/(2x^3) = 3/2. Thus, as x approaches positive infinity, f(x) approaches 3/2.