University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2022
Views:

86

Pages:

5
Author:

Quintin Ware

One to One Function 29/Oct/2021 A function f is called a one-to-one function (1-1) if f(x₁) = f(x₂) ⇒ x₁ = x₂ i.e. by Horizontal line test: A Function is (1-1) if no horizontal line intersects its graph more than once Example graph of a onetoone function quadratic function not one to one f(x₁) = f(x₂) 9 = 9 but 2 ≠ (-2) ∴ not (1-1) Thm, if f(x) 1-1 function ⇒ f(x) has inverse ∀ inverse function f⁻¹(x) Example: f(x) = 3x - 5 find f⁻¹(x): f(x) = 3x - 5 y = 3x - 5 5 + y = 3x x = (5 + y) / 3 f⁻¹(x) = (5 + x) / 3 **Note that:** 1. f: A → B * Dom f = A * Range f = B 2. f⁻¹: B → A * Dom f⁻¹ = Range f * Range f⁻¹ = Dom f **Example: Find the range of f(x) = (x-1)/(1+x)** Domain f(x): ℝ - {-1} Range f = Domain f⁻¹(x) f(x) = y ⇒ y = (x-1)/(1+x) ① y(1+x) = x-1 ② y + xy = x-1 ③ xy - x = -1 - y ④ x(y-1) = -1-y ⑤ x = (-1-y)/(y-1) Domain ℝ - {-1} ⇒ Range = ℝ - {-1} Example: f(x) = x³ + 4x + 1, find f⁻¹(6) f(2) = 6 ⇒ f⁻¹(6) = 2 ⇒ f⁻¹(x) = 1 + 4x + x³ + 6 1 + 4 + 1 = 6 Example 2: f(x) = x - 5x², x ≥ 1, find f⁻¹(x) y = x - 5x² y = -5x² + x y = -5(x² - x/5) y = -5(x² - x/5 + 1/100 - 1/100) y = -5(x - 1/10)² + 1/20 (y - 1/20)/-5 = (x - 1/10)² √((1/20 - y)/-5) = x - 1/10 1/10 + √((1/20 - y)/-5) = x f⁻¹(x) = 1/10 + √((1/20 - x)/-5)