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

2019
Views:

75

Pages:

5
Author:

Nicole Wang

**Chapter 1** **Mon, Oct 18** **Function & Models** [Image of two sets, A and B, with arrows connecting elements from A to B] * Each element in Domain attached to one element in the Range **Domain** **Range** **Definition:** f: A → B, f is a function from A to B "∀a ∈ A" to a unique element "b ∈ B" denoted by y = f(x) **Example:** [Image of two mappings, one representing a function and the other a relation] f(3) = 1 f(4) = 1 f(5) = 2 f is a function g(3) = 2 g(3) = 1 g is a relation Not a function X **Vertical line test (one-to-one)** [Image of three graphs: one passing the vertical line test, one failing, and one passing] (Function) (Relation) (Function) **Example:** [Image of a graph with a line segment connecting (2,2) to (5,3)] * **Domain:** [2, 5] * **Range:** [2, 3] **Domain:** The set of all possible (input) x **Range:** The set of all possible (output) y **Types of Functions:** 1. **Polynomials:** f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where n is a positive integer, a_n ≠ 0 2. **Rational Function:** f(x) = p(x) / q(x), where p(x), q(x) are polynomials, q(x) ≠ 0 3. **Absolute Function:** f(x) = |x| 4. **Exponential Function:** f(x) = a^x 5. **Logarithmic Function:** f(x) = log_a x 6. **Root Function:** f(x) = ⁿ√x 7. **Trigonometric Function** **Examples: Classify each function** 1. f(x) = 3x^8 - 7x^2 + x - 1 Poly: Degree of 8 2. f(x) = x^2 + 2x + 5 Poly: Quadratic function 3. f(x) = 2x + 1 Linear function 4. f(x) = x^3 - 2x^2 + 1 Cubic function 5. f(x) = (3x - 1) / (x^2 + 1) Rational function 6. f(x) = √(3x + 1) Root function 7. f(x) = |x - 1| Absolute value function **Even & Odd** 1. **A function f is said to be an even function** if f(-x) = f(x) **Examples:** * f(x) = x² (-x)² = x² = f(x) - even function * f(x) = |x| |-x| = |x| * f(x) = cos(x) cos(-x) = cos(x) [Image of a parabola symmetric about the y-axis] * Even functions are symmetric about the y-axis 2. **A function f is said to be an odd function** if f(-x) = -f(x) **Examples:** * f(x) = x³ (-x)³ = -x³ = -f(x) - odd function * f(x) = -x^5 - odd * f(x) = sin(x) sin(-x) = -sin(x)