1.04E - Rational Rule and Inverse | Lecture Note

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2024
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1.04 Rational Functions 1.04E FINDING RULE / INVERSE Rational Functions • Unless you are explicitly given parameter 𝑏 𝑎 𝑓 𝑥 = +𝑘 𝑥−ℎ Simplified Standard Form • Asymptotes: Horizontal 𝑦 = 𝑘 Vertical 𝑥 = ℎ • Intersection of Asymptotes (ℎ, 𝑘) • Domain 𝑑𝑜𝑚 𝑓 = ℝ\{ℎ} • Range 𝑟𝑎𝑛 𝑓 = ℝ\{𝑘} Key Information 𝑥 = −2 • Asymptotes: ℎ = −2 𝑘=3 5 − ,2 2 • Point: • 2= 𝑎 5 −2+2 +3 −1 = −1 = 1 2 5 − ,2 2 𝑦=3 𝑎 5 −2+2 𝑎 −0.5 =𝑎 Example 𝑓 11 2 𝑥𝑓 = 𝑥 = + 3+ 3 2(𝑥 𝑥 ++ 2 2) Find the rule of the rational function that is undefined over 𝑥 = −2, has a range of 𝑦 ∈ ℝ 𝑦 ≠ −7 and passes through the point 5, −3 We Know: ℎ = −2 𝑘 = −7 𝑃 5, −3 28 ∴𝑓 𝑥 = −7 𝑥+2 Example 2 𝑎 𝑓 𝑥 = −7 𝑥+2 𝑎 −3 = −7 5+2 𝑎 4= 7 28 = 𝑎 1. Identify given parameter: 2. Create two rules using parameter and two points 3. Solve system of equations Given parameter and 2 points • Find the rule of the rational function that passes through 11 15 1, and 5, over the domain of ℝ\{2} 2 2 We Know: ℎ=2 𝑓 𝑥 = 𝑓 𝑥 = 𝑎 +𝑘 𝑥−2 3 + 7, 𝑥 ≠ 2 2(𝑥 − 2) 11 2 11 𝑎 = +𝑘 2 1−2 15 2 15 𝑎 = +𝑘 2 5−2 11 𝑎 = +𝑘 2 −1 15 𝑎 = +𝑘 2 3 11 +𝑎 =𝑘 2 15 1 − 𝑎=𝑘 2 3 1, 5, 11 15 1 +𝑎 = − 𝑎 2 2 3 Example 3 11 3 𝑘= + =7 2 2 4 𝑎=2 3 3 𝑎= 2 15 1 3 𝑘= − =7 2 3 2 A Quotient (ratio) of two linear functions: 1. Find the rules that correspond to the linear functions 2. ‘Stack’ them according to the situation 3. You will have general Form As Quotient of 2 Linears • To enter a basketball tournament, players must pay their share of the $500 transportation cost, plus an additional $20 fee. Write a rule that will determine the amount each player will pay based on the number of players attending. 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 = # 𝑜𝑓 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 • Total Cost: $20 per person + $500 transport: 500 + 20𝑥 • Total players: 𝑥 • Rule: 𝑦= Example 4 500+20𝑥 𝑥 • 10g of substance A are placed in 100mL of water. Every minute 2g of substance A is added and 5mL of water is added. Write a rule to determine the concentration of substance A over time, in minutes (that is, the amount of substance A compared to the amount of water). 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝐴 𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓𝐴 = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 • Amount of A: 10g + 2g per min: 10 + 2𝑥 • Amount of water: 100mL + 5ml per min: 100 + 5𝑥 • Rule: y= Example 5 10+2𝑥 100+5𝑥 •The inverse of a rational function is also a rational function Inverse • 𝑓 𝑥 = • 𝑓 −1 3 + 𝑥−2 1, 𝑥 ≠ 2 𝑥 ⇒𝑥= 𝑓 −1 3 𝑦−2 +1 3 𝑥−1= 𝑦−2 3 𝑦−2= 𝑥−1 3 𝑥 = + 2, 𝑥 ≠ 1 𝑥−1 Example 1 𝒇 𝒙 = 𝒇−𝟏 𝟑 𝒙−𝟐 + 𝟏, 𝒙 ≠ 𝟐 𝟑 𝒙 = + 𝟐, 𝒙 ≠ 𝟏 𝒙−𝟏 Mirror on 𝒚 = 𝒙 Find the Inverse 𝑎 and 𝑏 stay the same ℎ and 𝑘 switch!! Standard Form: Inverse • 𝑓 𝑥 = 4𝑥+3 𝑥+7 𝑥 ≠ −7 4𝑦 + 3 𝑥= 𝑦+7 𝑥𝑦 + 7𝑥 = 4𝑦 + 3 𝑥𝑦 − 4𝑦 = −7𝑥 + 3 𝑦 𝑥 − 4 = −7𝑥 + 3 𝑓 −1 −7𝑥 + 3 𝑥 = ;𝑥 ≠ 4 𝑥−4 Example 2 𝑐𝑜𝑚𝑝𝑎𝑟𝑒 𝑓 −1 𝑥 𝑓 −1 𝑥 = −7𝑥+3 ,𝑥 𝑥−4 ≠4 Example 2 𝑡𝑜 𝑓(𝑥) 𝑓 𝑥 = 4𝑥+3 𝑥+7 a1, b2 switch places & change sign, a2, b1 stay same! General Form: Inverse •Visions 1 •P67: #3ace •P68: #5a(1,3,5)