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Domain: a set of first elements in a relation (all of the x values). This is the input. Range: The second elements in a relation (all of the y values). This is the output. How would you use your calculator to solve 52? Input 5 Output x2 25 The number you entered is the input number (or x-value on a graph). The result is the output number (or yvalue on a graph). A function is a relation that gives a single output number for every valid input number (x values cannot be repeated). A relation is a rule that produces one or more output numbers for every valid input number (x and y values may be repeated). There are many ways to represent relations: Graph Table of values A set of ordered pairs Mapping Equations These are all ways of showing a relationship between two variables. Function X values are always located on the left and y values are on the right. They can be represented by words, symbols or numbers. This represents a function as every input value (x) has only been used once. A Relation is a rule that produces one or more output numbers for every valid input number (x and y values may be repeated). This represents only a relation because the input value or x-value of 2 was used twice. Therefore this relation is not a Function. All functions are relations but not all relations are functions! Vertical Line Test: 1.Draw a vertical line through the graph. 2. See how many times the vertical line intersects the graph. 3. Only Once – Pass (function) More than Once – Fail (not function) To determine if a graph is a function, we perform the vertical line test. -- Yes, it is a function. -- No, it is not a function. Is this graph a function? Only crosses at one point. Yes, this is a function because it passes the vertical line test. Is this graph a function? Crosses at more than one point. No, this is not a function because it does not pass the vertical line test. Is this graph a function? Only crosses at one point. Yes, this is a function because it passes the vertical line test. To determine if a table is a function, we look at the x column (domain). If each number in the x column appears only once in that column, it is a function. Also, each number in the x column corresponds with one and only one value in the y column. FUNCTIONS Many to One Relationship One to One Relationship X 1 2 3 Y 0 0 0 X 1 2 3 Y 5 7 9 Is this relation a function? X 1 2 3 4 Y 5 4 3 2 Every number just appears once. Yes, this is a function because each number in the column only appears once. x Is this relation a function? X 7 9 10 10 Y 7 9 8 10 The number 10 appears more than once. No, this is not a function because 10 appears in the x column more than once. Is this relation a function? X 1 2 3 4 Y 0 0 0 0 Every number just appears once. Yes, this is a function because each number in the column only appears once. x Is this relation a function? X Y 0 1 2 2 4 3 2 4 The number 2 appears more than once. No, this is not a function because 2 appears in the x column more than once. How to Identify Functions and Relations from Ordered Pairs List of Ordered Pairs {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} Look at the X’s (numbers on the left) List them out. 2, -1, 4, 9, 0 If no X values are repeated it is a function. If X values are repeated it is a relation. Identifying Functions with Equations