Tips Use parentheses ( ) if the start or end of an interval is not included in the domain or range. o This happens when there is an open circle at a p

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1 SM 2 Date: Section: Objective: Domain: The set of all (the x-values) of a relation. If a relation is represented by a graph, the domain is the set of all of points on the graph. You can think of it as the graph s shadow on the x-axis. Range: The set of all (the y-values) of a relation. If a relation is represented by a graph, the range is the set of all of points on the graph. You can think of it as the graph s shadow on the y-axis. If the graph is a set of unconnected points, the domain and range are just lists of the x and y coordinates, respectively. However, if the graphs contain connected points, they contain an infinite number of points, so we can t list the coordinates. One way we solve this is to use interval notation. Interval Notation Domain and range are often written in interval notation. The numbers are where the interval starts and stops. If an endpoint is included, put it in a bracket [ ]. If an endpoint is not included, put it in parentheses ( ). If the interval goes on forever, use or. These always get put in parentheses ( ). Multiple intervals are connected with the union sign, which is the math symbol for or. Graph Interval Notation Set-Builder Notation Meaning R All real numbers (, 3] Everything less than or equal to 3 { x x > 4} Everything greater than 4 ( 3,7 ) [ 2,1) (,1) [ 4, ) { x 5 x 8} { x x 5 and x 8} { x x 1 or x 2} Everything between 3 and 7, not including either 3 or 7 Everything between 5 and 8, including both 5 and 8 Everything between 2 and 1, including 2, but not including 1 Everything that is either less than or equal to 1 or greater than or equal to 2 Everything that is either less than 1 or greater than or equal to 4

2 Tips Use parentheses ( ) if the start or end of an interval is not included in the domain or range. o This happens when there is an open circle at a point or an asymptote (a line that the graph gets really close to, but never actually touches). Use brackets [ ] if the start or end of an interval is included in the domain or range. o It there is a point there (endpoint or vertex), use a bracket. Always use parentheses around and. Read the domain from left to right and the range from down to up. o Write the lower value or first and the higher value or last. Vertical Line Test If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function. The graph at left is not a function because one x-value has 3 different y-values. Examples: Determine whether each graph is the graph of a function. Then state the domain and range. a) b) c) d) e) f)

3 g) h) i) Domain and Range in Real Life Situations: In real life situations, it s important to think through what values make sense in the problem. You also need to think about which variable is the input and which is the output. In real-life problems, the domain is the values of the x variable (input) that make sense in the problem and the range is the possible values of the y variable (output). Examples: Find the real world domain and range for each situation. a) A ball is dropped from a window that is 64 feet above the ground. The ball takes 2 seconds to hit the ground. What are the domain and range if the height is a function of time? Circle which variable represents the domain: Time or Height Domain: Circle which variable represents the range: Time or Height Range: b) You are getting ready for the Homecoming dance. Your dad is going to let you borrow his new car, but you need to wash and fill it. The car wash costs $5 and the gas costs $3.89 per gallon. The car can hold 15 gallons of gas. What are the domain and range if the total cost is a function of the number of gallons? Circle which variable represents the domain: Cost or Number of Gallons Domain: Circle which variable represents the range: Range: Cost or Number of Gallons

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5 2 Examples: Find each value if f ( x) = x 2x + 3, g ( x) = 3x 5, and h( x). answers as simplified fractions, if necessary. Show all your work. a) f ( 2) b) g ( 1) c) h ( 4) x = Leave your 4 2x d) 2 g 3 e) f ( 5) f) h( 3)