2.3 – Functions and Relations Relations, Domain and Range Defn: A relation is a set of ordered pairs. A 0,0, 1,1, 1,1, 4,2, 4,2 Domain: The values of the 1st component of the ordered pair. domain A : 0, 1, 4 Range: The values of the 2nd component of the ordered pair. range A : 2, 1, 0, 1, 2 2.3 – Functions and Relations Relations, Domain and Range State the domain and range of each relation. x y x y 1 3 4 2 2 5 -3 8 -4 6 6 1 1 4 -1 9 3 3 5 6 domain : 4, 1, 2, 3 range : 3, 4, 5, 6 domain : 3, 1, 4, 5, 6 range : 1, 2, 6, 8, 9 2.3 – Functions and Relations Functions Defn: A function is a relation where every x value has one and only one value of y assigned to it. State whether or not the following relations could be a function or not. x y x y x y 4 2 1 3 2 3 -3 8 2 5 5 7 6 1 -4 6 3 8 -1 9 1 4 -2 -5 5 6 3 3 8 7 function not a function function 2.3 – Functions and Relations Functions State whether or not the following mappings of two relations could be a function or not. x y x y Relation A is not a function. Relation B is a function. For x = 1, there are two y values (3 and 4). For each value of x, there is one value of y. 2.3 – Functions and Relations Functions Graphs can be used to determine if a relation is a function. Vertical Line Test If a vertical line can be drawn so that it intersects a graph of an equation more than once, then the equation is not a function. 2.3 – Functions and Relations Functions The Vertical Line Test y y 2x 3 x y 0 -3 5 7 -2 -7 4 5 3 3 function x 2.3 – Functions and Relations Functions The Vertical Line Test y x y2 x y 1 1 1 -1 4 2 4 -2 0 0 not a function x 2.3 – Functions and Relations Functions Domain and Range from Graphs y Find the domain and range of the function graphed to the right. Use interval notation. Domain x Domain: [–3, 4] Range: [–4, 2] Range 2.3 – Functions and Relations Functions Domain and Range from Graphs Find the domain and range of the function graphed to the right. Use interval notation. y Range x Domain: (– , ) Range: [– 2, ) Domain 2.3 – Functions and Relations Function Notation Shorthand for stating that an equation is a function. Defines the independent variable (usually x) and the dependent variable (usually y). y 3x 1 y x 3x 1 y yx f x f x 3x 1 2.3 – Functions and Relations Function Notation Function notation also defines the value of x that is to be use to calculate the corresponding value of y. f x 2 x 5 find f(3). f 3 23 5 f 3 1 3, 1 g(x) = x2 – 2x find g(–3). f(x) = 4x – 1 find f(2). f(2) = 4(2) – 1 g(–3) = (-3)2 – 2(-3) f(2) = 8 – 1 f(2) = 7 (2, 7) g(–3) = 9 + 6 g(–3) = 15 (–3, 15) 2.3 – Functions and Relations Function Notation Given the graph of the following function, find each function value by inspecting the graph. y f(x) f(5) = 7 ● f(4) = 3 f(5) = 1 f(6) = 6 ● ● ● x 2.4 – linear Equations in Two Variables and Linear Functions Three Forms of an Equation of a Line Slope-Intercept Form: This form is useful for graphing, as the slope and the y-intercept are readily visible. Point-Slope Form: The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line. Standard Form: Note: A, B, and C cannot be fractions or decimals. 2.4 – linear Equations in Two Variables and Linear Functions Slope of a Line y y2 y1 Slope formula = m x x2 x1 Y Y Y X X X Green slant line: positive slope Vertical line: undefined or no slope Horizontal line: slope = 0 Red slant line: positive slope Δ𝑥 = 0 Δ𝑦 = 0 Linear Functions Constant Functions 2.4 – linear Equations in Two Variables and Linear Functions Slope of a line y y2 y1 Slope formula = m x x2 x1 Find the slope of the line containing the following points: 7−4 𝑚= −3 − 1 𝑚= 3 3 =− −4 4 −3, 7 𝑎𝑛𝑑 1, 4 2.4 – linear Equations in Two Variables and Linear Functions Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 𝑚: 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑏: 𝑡ℎ𝑒 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 0, 𝑏 8 𝑦= 𝑥−6 11 5𝑥 + 2𝑦 = 7 8 𝑚= 11 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0, −6 2𝑦 = −5𝑥 + 7 5 7 𝑦=− 𝑥+ 2 2 5 𝑚=− 2 7 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0, 2 2.4 – linear Equations in Two Variables and Linear Functions The Average Rate of Change If f is defined on the interval [x1, x2], then the average rate of change of f on the interval [x1, x2] is the slope of the secant line containing (x1, f(x1)) and (x2, f(x2)). Average rate of change: f ( x2 ) f ( x1 ) y y2 y1 m m x2 x1 x x2 x1 2.4 – linear Equations in Two Variables and Linear Functions Determine the average rate of change From (–1.5, 0) to (0, 2) 𝑓 0 − 𝑓 −1.5 𝑚= 0 − −1.5 2 𝑚= 1.5 2−0 𝑚= 0 − −1.5 4 𝑚= 3 From (0, 2) to ( 2.5, 1) 𝑓 2.5 − 𝑓 0 𝑚= 2.5 − 0 −1 𝑚= 2.5 2 𝑚=− 5 1−2 𝑚= 2.5 − 0 2.4 – linear Equations in Two Variables and Linear Functions Solving Equations and Inequalities Graphically 𝑦 = 𝑥 + 5 𝑎𝑛𝑑 𝑦 = −3𝑥 − 3 • 𝑥 + 5 = −3𝑥 − 3 4𝑥 + 5 = −3 4𝑥 = −8 𝑥 = −2 𝑥 + 5 > −3𝑥 − 3 −2, ∞ 𝑥 + 5 ≤ −3𝑥 − 3 −∞, −2 2.4 – linear Equations in Two Variables and Linear Functions Solving Equations and Inequalities Graphically 𝑦 = −4𝑥 + 4 𝑎𝑛𝑑 𝑦 = 2𝑥 − 2 −4𝑥 + 4 = 2𝑥 − 2 −6𝑥 + 4 = −2 • −6𝑥 = −6 𝑥=1 −4𝑥 + 4 ≥ 2𝑥 − 2 −∞, 1 −4𝑥 + 4 ≤ 2𝑥 − 2 1, ∞