Rational Functions

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Rational Functions · Grade 12 · Worksheet 1

1. 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

2. Consider the rational function f(x) = (3x³ - 2x² + 5x - 1)/(2x³ + 4x - 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: 1.5 Solution: To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and denominator. f(x) = (3x³ - 2x² + 5x - 1) / (2x³ + 4x - 7) Identify the degrees of numerator and denominator.
   Full step-by-step solution

   To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and denominator. The function is: f(x) = (3x³ - 2x² + 5x - 1) / (2x³ + 4x - 7) Step 1: Identify the degrees of numerator and denominator. - The numerator is 3x³ - 2x² + 5x - 1. The highest power is x³, so degree = 3. - The denominator is 2x³ + 4x - 7. The highest power is x³, so degree = 3. Step 2: Apply the rule for horizontal asymptotes when degrees are equal. When the degrees of numerator and denominator are equal, the horizontal asymptote is: y = (leading coefficient of numerator) / (leading coefficient of denominator) Step 3: Identify the leading coefficients. - Leading coefficient of numerator (coefficient of x³) = 3 - Leading coefficient of denominator (coefficient of x³) = 2 Step 4: Calculate the horizontal asymptote. y = 3 / 2 = 1.5 Therefore, the horizontal asymptote is the horizontal line y = 1.5. Final answer: 1.5

3. lim(x→∞) (5x⁴ - 2x³ + 7x - 1)/(3x⁴ + 4x² - 9) = ? Answer: 5/3 Solution: Identify the degrees of the numerator and denominator. Both have degree 4. Since the degrees are equal, the limit is the ratio of the leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. Both have degree 4. Step 2: Since the degrees are equal, the limit is 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 3. Step 5: The limit is 5/3. The answer is 5/3.

4. Olivia is a water quality analyst studying the concentration of a chemical in a river downstream from a factory. The concentration C(x) in parts per million (ppm) is modeled by the rational function C(x) = (5x² + 20x) / (x² - 25), where x represents the distance downstream in kilometers from the discharge point. Olivia needs to predict the long-term concentration as the distance becomes very large and identify any distances where the model breaks down because the concentration is undefined. What are the horizontal asymptote and vertical asymptotes of this concentration function? Additionally, describe the behavior of C(x) near the vertical asymptote at the positive distance. Answer: Horizontal asymptote: y = 5; Vertical asymptotes: x = 5 and x = -5; Near x = 5, as x approaches 5 from the left, C(x) approaches negative infinity, and as x approaches 5 from the right, C(x) approaches positive infinity. Solution: Find vertical asymptotes by setting the denominator equal to zero: x² - 25 = 0 → x² = 25 → x = 5 or x = -5.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero: x² - 25 = 0 → x² = 25 → x = 5 or x = -5. Check that the numerator is not zero at these points: at x = 5, numerator = 5(5)² + 20(5) = 125 + 100 = 225 ≠ 0; at x = -5, numerator = 5(-5)² + 20(-5) = 125 - 100 = 25 ≠ 0. So vertical asymptotes at x = 5 and x = -5. Step 2: Find the horizontal asymptote by comparing degrees: both numerator and denominator are degree 2. The leading coefficient of the numerator is 5, and the denominator is 1. So the horizontal asymptote is y = 5/1 = 5. Step 3: Describe behavior near x = 5. Factor numerator: 5x² + 20x = 5x(x + 4). Denominator: (x - 5)(x + 5). Near x = 5, the factor (x - 5) changes sign. For x slightly less than 5 (e.g., x = 4.9), numerator is positive (5*4.9*8.9 > 0), denominator (4.9-5) is negative and (4.9+5) is positive, so overall negative → C(x) → -∞. For x slightly greater than 5 (e.g., x = 5.1), numerator is positive, denominator (5.1-5) is positive and (5.1+5) is positive, so overall positive → C(x) → +∞. The answer is: Horizontal asymptote y = 5; Vertical asymptotes x = 5 and x = -5; Near x = 5, C(x) → -∞ from the left and +∞ from the right.

5. 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 - 12)/(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 concentration level and identify any time points where the concentration becomes undefined. What is the horizontal asymptote of this function, and at what time value does the vertical asymptote occur? Answer: Horizontal asymptote: y = 3; Vertical asymptote: t = 2 Solution: The horizontal asymptote represents the long-term behavior of the function as the input grows very large, while vertical asymptotes indicate values where the function is undefined due to division by zero.
   Full step-by-step solution

   Rational functions often model real-world scenarios where quantities approach limiting values or become undefined at certain points. The horizontal asymptote represents the long-term behavior of the function as the input grows very large, while vertical asymptotes indicate values where the function is undefined due to division by zero. When analyzing such functions, it's important to consider both the algebraic structure and the practical interpretation within the given context.

6. A rational function is graphed on a coordinate plane with vertical asymptotes at x = -1 and x = 4, and a horizontal asymptote at y = 2. The function passes through the point (2, 3). Determine the equation of this rational function in the form f(x) = (ax + b)/((x + 1)(x - 4)). Answer: f(x) = (4x - 2)/((x + 1)(x - 4)) Solution: The vertical asymptotes at x = -1 and x = 4 indicate factors of (x + 1) and (x - 4) in the denominator.
   Full step-by-step solution

   Step 1: The vertical asymptotes at x = -1 and x = 4 indicate factors of (x + 1) and (x - 4) in the denominator. Step 2: The horizontal asymptote at y = 2 suggests that the degrees of numerator and denominator are equal, and the ratio of leading coefficients is 2. Step 3: Since the denominator is (x + 1)(x - 4) = x^2 - 3x - 4 (degree 2), the numerator must also be degree 2 or less. With the given form f(x) = (ax + b)/((x + 1)(x - 4)), the numerator is degree 1, so for the horizontal asymptote to exist, we consider the limit as x approaches infinity: lim(x→∞) (ax + b)/(x^2 - 3x - 4) = 0, which contradicts the horizontal asymptote at y = 2. Therefore, the numerator must actually be degree 2. Step 4: Let's use the correct form: f(x) = (ax^2 + bx + c)/((x + 1)(x - 4)). For the horizontal asymptote to be y = 2, the degrees must be equal and a/1 = 2, so a = 2. Step 5: Now we have f(x) = (2x^2 + bx + c)/((x + 1)(x - 4)). Step 6: Use the point (2, 3): f(2) = (2(2)^2 + b(2) + c)/((2 + 1)(2 - 4)) = (8 + 2b + c)/((3)(-2)) = (8 + 2b + c)/(-6) = 3 Step 7: So (8 + 2b + c)/(-6) = 3, which means 8 + 2b + c = -18, so 2b + c = -26. Step 8: We need another condition. Since no other points are given, let's assume the simplest case where the numerator and denominator have no common factors (no holes). We can choose a specific value for b or c. Let's set b = 0 to find c. Step 9: If b = 0, then c = -26. So f(x) = (2x^2 - 26)/((x + 1)(x - 4)). Step 10: Check if this satisfies all conditions: Vertical asymptotes at x = -1 and x = 4 ✓, Horizontal asymptote at y = 2 ✓, Passes through (2, 3): f(2) = (2(4) - 26)/((3)(-2)) = (8 - 26)/(-6) = (-18)/(-6) = 3 ✓ The equation is f(x) = (2x^2 - 26)/((x + 1)(x - 4)).