Lesson 1 – Characteristics & Properties of Functions Warm-up: Prerequisite Skills Relation: A relationship between two variables, values of the independent variable are with values of the dependent variable. Relations may be described in various ways: 1. LISTING the elements. 2. TABLE OF VALUES Ex. { (1,2), (3,4), (5,6), (7,8)} x y = 3x + 1 –1 –2 0 1 1 4 2 7 3. GRAPHICALLY 4. MAPPING DIAGRAM 𝑥 𝑦 0 –4 –2 0 2 4 2 4 5. EQUATIONS ex. 𝑦 = 2𝑥 − 5 DOMAIN: The set of all values for which the independent variable is defined (usually x). RANGE: The set of all values for which the dependent variable is defined (usually y). Example State the domain and range for each of the following: a) b) x y 5 2 5 4 5 6 5 8 D= R= D= R= c) { (1,3), (2, -4), (3,5), (4,5)} d) x y D= 0 R= 2 4 D= R= –4 –2 0 2 4 Function: A relation in which there is only one value of the dependent variable for each value of the independent variable. THE DOMAIN CANNOT REPEAT!! If each vertical line drawn intersects the graph of a relation in at most one point, then the relation is a function. Vertical Line Test (VLT) Example 2 For each of the following relations, determine the domain and range using real numbers. State whether or not the relation is a function. a) b) c) D= D= D= R= R= R= Function? Function? Function? FUNCTION NOTATION: Example a) f(–3) f(x) is read “the value of f at x” or “f of x”. It is used to represent the value of the dependent variable (y) for a given value of the independent variable (x). (y and f(x) are the same!) If f(x) = 3x2 – 4, determine the value of b) f(½)