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Name______________________ Date_______________________ Period______________________ Relations and Functions Domain and Range Algebra II Honors I. Relations A relation is any set of _________________. The set of all first components of the ordered pairs is called the __________of the relation, and the set of all second components is called the ___________of the relation. Example: Find the domain and range of the relation: {(1994,56.21), (1995,51.00), (1996,47.70), (1997, 42.78), (1998, 39.13)} II. Functions A function is a ___________ where each element in the domain corresponds to _______________ element in the range. Note: A function is a relation in which _____________________________ ____________________________________________________________. Example: Determine if the following relations are functions: a. {(1,6),(2,6),(3,8),(4,9)} Example: Determine whether each relation is a function. S = {(1,2), (3,4), (5,6), (7,8)} III. b. {(6,1),(6,2),(8,3),(9,4)} T= {(1,2), (3,4), (6,5), (1,5)} Expressing Functions Relations and functions can be expressed as sets of ordered pairs, graphs, and equations. In an equation, the variable, x is ________________ and the variable y is ________________. If the exponent on the y is even, the equation will not define y as a function of x. It will result in more than one value for the y to go with the x value. Name______________________ Date_______________________ Period______________________ IV. Function Notation When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function's inputs and the range as the set of the function's outputs. The input is represented by x and the output by f(x). The special notation f(x), read "f of x" or "f at x" represents the value of the function at the number x. If a function is named f and x represents the independent variable, the notation f(x) corresponds to the y-value for a given x. Thus, f(x) = 4 - x2 and y = 4 - x2 define the same function. This function may be written as y = f(x) = 4 - x2 Example: If f(x) = x2 + 3x + 5, evaluate: V. a. f(2) Obtaining Information From Graphs You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows. A closed dot indicates that the graph ________ _______ extend beyond this point and the point belongs to the graph. This would _________ the endpoint. An open dot indicates that the graph does not extend beyond this point and the point ______ _____ belong to the graph. This would __________ the endpoint. An arrow indicates that the graph extends _______________ in the direction in which the arrow points. Example: Use the graph of the function f to answer the following questions. a.) What are the function values f(-1) and f(1)? b.) What is the domain if f(x)? c.) What is the range of f(x)? Name______________________ Date_______________________ Period______________________ VI. The Vertical Line Test For Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Example: Problems: Evaluate each function for the given values. 1. f(x) = 3x + 7 f(4) = 2. f(x) = -6 + f(16) =