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

2020
Views:

462

Pages:

13
Author:

Apolenajz

**Domain & Range** **Oct 20, 2021** **1. Linear Function** y = ax + b a = slope Domain = ℝ (all real numbers) Range = ℝ (all real numbers) [Image of a linear function graph] **2. Cubic Function** Domain = ℝ Range = ℝ [Image of a cubic function graph] **3. Quadratic Function** e.g.) P(x) = x² Domain = ℝ Range = [0, ∞) [Image of a parabola opening upwards] P(x) = -x² Domain = ℝ Range = (-∞, 0] [Image of a parabola opening downwards] P(x) = x² + 5 Domain = ℝ Range = [5, ∞) [Image of a parabola opening upwards, shifted up by 5 units] Note: x² + a Range = [a, ∞) **f(x) = -x² + 3** Domain = ℝ Range = (-∞, 3] [Image of a parabola opening downwards, with vertex at (0,3)] **f(x) = (x+2)²** x+2 = 0 x = -2 Domain = ℝ Range = [0, ∞) [Image of a parabola opening upwards, with vertex at (-2,0)] **f(x) = (x-2)²** Domain = ℝ Range = [0, ∞) [Image of a parabola opening upwards, with vertex at (2,0)] **f(x) = (x-2)² + 3** Domain = ℝ Range = [3, ∞) [Image of a parabola opening upwards, with vertex at (2,3)] **New function from old** **Domain Rules:** 1. Domain f(g(x)) = Dom f(x) ∩ Dom g(x) 2. Domain (f/g)(x) = Dom f(x) ∩ Dom g(x) - {x: g(x) = 0} 3. Domain f(g(x)) = {Dom f(x) ∩ Dom g(x)} - {x: g(x) = 0} **Example: Find Domain** 1. f(x) = 7 / √(x-2) x - 2 ≥ 0 x ≥ 2 ℝ ∩ [2, ∞) = [2, ∞) 2. f(x) = √(x+2) + √(3-x) x + 2 ≥ 0 and 3 - x ≥ 0 x ≥ -2 and x ≤ 3 (-∞, 3] ∩ [-2, ∞) = [-2, 3] **6. f(x) = √x** x ≥ 0 → [0, ∞) **7. f(x) = 1 / (1 + √(1-x²))** Domain: [-1, 1] Range: [1/2, 1] **8. f(x) = (x-1)² / (x-1)** x - 1 ≠ 0 x ≠ 1 ℝ - {x=1} (-∞, 1) U (1, ∞) **9. f(x) = x / |x-1|** x - 1 ≠ 0 x ≠ 1 ℝ except {2, 0} x - 1 = -1 or x - 1 = 1 x = 0 or x = 2