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Lesson 1.2, pg. 138 Functions & Graphs Objectives: To identify relations and functions, evaluate functions, find the domain and range of functions, determine whether a graph is a function, and graph a function. Domain & Range • A relation is a set of ordered pairs. • Domain: first components in the relation (independent); x-values • Range: second components in the relation (dependent, the value depends on what the domain value is); y-values • Find the domain and range of the relation. {(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )} FUNCTIONS • Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element. Every “x” has only one “y”. Mapping – illustrates how each member of the domain is paired with each member of the range (Note: List domain and range values once each, in order.) Draw a mapping for the following. (5, 1), (7, 2), (4, -9), (0, 2) x y 0 4 5 7 -9 1 2 Is this relation a function? See Example 2, page 150. Determine whether each relation is a function: A) {(1,2), (3,4), (5,6), (5,8)} B) {(1,2), (3,4), (6,5), (8,5)} Functions as Equations Determine whether the equation defines y as a function of x. a) x2 y 4 b) x2 y 2 4 1. Solve for y in terms of x. 2. If two or more values of y can be obtained for a given x, the equation is not a function. Determine if the equation defines y as a function of x. A) 2x + y = 6 B) x2 + y2 = 1 C) x2 + 2y = 10 Evaluating a Function • Common notation: f(x) = function • Evaluate the function at various values of x, represented as: f(a), f(b), etc. • Example: f(x) = 3x – 7 f(2) = f(3 – x) = If f(x) = x2 – 2x + 7, evaluate each of the following. • a) f(-5) b) f(x + 4) c) f(-x) See Example 4, page 143 for additional practice. Determine if a relation is a function from the graph? • Remember: to be a function, an x-value is assigned ONLY one y-value . • On a graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line. • THUS, the vertical line test! If a vertical line drawn on any part of your graph touches more than one point, it is NOT the graph of a function. Graphs of Functions Step 1: Graph the relation. (Use graphing calculator or pencil and paper.) Step 2: Use the vertical line test to see if the relation is a function. • Vertical line test – If any vertical line passes through more than one point of the graph, the relation is not a function. Determine if the graph is a function. a) y b) 5 y x x -5 -5 5 Here’s more practice. c) y y d) x x Example Analyze the graph. y f ( x) x 2 3x 4 a. Is this a function? b. Find f(4) c. Find f(1) d. For what value of x is f(x)=-4 y Find f(7). (a) 0 (b) 1 x (c) 1 (d) 2 Can you identify domain & range from the graph? • Look horizontally. What x-values are contained in the graph? That’s your domain! • Look vertically. What y-values are contained in the graph? That’s your range! • Write domain and range using interval or set-builder notation. • See Example 8, page 148. Domain: set of all values of x Range: set of all values of y • Always write the domain and range in interval notation when reading the domain and range from a graph. • Use brackets [ or ] to show values that are included in the graph. • Use parentheses ( or ) to show values that are NOT included in the graph. Identify the function's domain and range from the graph y y Domain (-1,4] Range [1,3) x Domain [3,) Range [0,) Example y Identify the Domain and Range from the graph. Example y Identify the Domain and Range from the graph. x Example y Identify the Domain and Range from the graph. x Find the Domain and Range. y D:(-, ) R:(-5,7] (b) D:(-5,) R: (-, ) (c) D:(-, ) R: [-5,) (d) D:[-, ] R: [-5,] (a) x What is the difference in the two sets below, and when should we use each to describe the domain of a function? {1,2,3,4} [1,4] Finding intercepts: • x-intercept: where the function crosses the xaxis. What is true of every point on the x-axis? The y-value is ALWAYS zero. • y-intercept: where the function crosses the yaxis. What is true of every point on the yaxis? The x-value is ALWAYS zero. • Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin! We can identify x and y intercepts from a function's graph. To find the x-intercepts, look for the points at which the graph crosses the x axis. The y-intercepts are the points where the graph crosses the y axis. The zeros of a function, f, are the x values for which f(x)=0. These are the x intercepts. By definition of a function, for each value of x we can have at most one value for y. What does this mean in terms of intercepts? A function can have more than one x-intercept but at most one y intercept. y Example Find the x intercept(s). Find f(-4) x Example y Find the x and y intercepts. Find f(5). x Summary • Domain = x values • Range = y values • Use the vertical line test to verify if a graph is a function. • To evaluate means to substitute and simplify. • Intercepts – where function crosses the x-or yaxis