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This lesson introduces the key concepts of functions, including domain and range, one-to-one and onto functions, and discrete and continuous functions. Examples and practice problems are provided to reinforce understanding.

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  1. Five-Minute Check (over Chapter 1) Then/Now New Vocabulary Key Concept: Functions Example 1: Domain and Range Key Concept: Vertical Line Test Example 2: Graph a Relation Example 3: Evaluate a Function Example 4: Real-World Example: Discrete and Continuous Functions Example 5: Choose the Correct Model Lesson Menu

  2. You identified domains and ranges for given situations. • Determine whether functions are one-to-one and/or onto. • Determine whether functions are discrete or continuous. Then/Now

  3. one-to-one function • onto function • discrete relation • continuous relation • vertical line test • independent variable • dependent variable • function notation • codomain Vocabulary

  4. Key Concept

  5. Domain and Range State the domain and range of the relation. Then determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. Example 1

  6. Domain and Range D = {–4, –3, 0, 1, 3} The domain consists of the x- values in the points of the relation. R = {–2, 1, 2, 3} The range consists of the y- values in the points of the relation. Relation is a function. Each member of the domain is paired with one member of the range. Not one-to-one. Each element of the domain does not pair with exactly one element of the range. Example 1

  7. Domain and Range Answer: D = {‒4, ‒3, 0, 1, 3}; R = {‒2, 1, 2, 3}Each member of the domain is paired with one member of the range, so this relation is a function. It is onto, but not one-to-one. Example 1

  8. Key Concept

  9. Graph a Relation Graph y = 3x – 1 and determine the domain and range. Then determine whether the equation is a function, is one-to-one, onto, both, or neither. State whether it is discrete or continuous. Example 2

  10. Graph a Relation m = 3 and b = –1 Graph the equation and use the vertical line test to determine if the equation is a function. D = All real numbers The graph goes from −∞ to ∞ along the x- axis. R = All real numbers The graph goes from −∞ to along the y- axis. Continuous There are no interruptions in the graph. Example 2

  11. Graph a Relation One-to-one and onto Each element of the domain is paired with one element of the range and each element of the range is paired with one element of the domain. Answer: The domain is all real numbers; the range is all real numbers; the equation is a function; the function is both one-to-one and onto; the equation is continuous. Example 2

  12. Evaluate a Function Given f (x) = x3 – 3, find each value. A.f (–2) f (x) = x3 – 3 Original function. f(–2) = (–2)3 – 3Substitute f(–2) = –8 – 3Evaluate (–2)3 f(–2) = –11Simplify Answer: –11 Example 3A

  13. Evaluate a Function Given f (x) = x3 – 3, find each value. B.f (2t) f(x) = x3 – 3Original function. f(2t) = (2t)3 – 3Substitute f(2t) = 8t3 – 3(2t)3 = 8t3 f(2t) = 8t3 – 3Simplify Answer: 8t3 – 3 Example 3B

  14. Discrete and Continuous Functions TRANSPORTATION The table shows the average fuel efficiency in miles per gallon for SUVs for several years. Graph this information and determine whether it represents a function. Is the relation discrete or continuous? Real-World Example 4

  15. Discrete and Continuous Functions Graph the data in the table. Label the horizontal axis with the year and the vertical axis miles/gallon. Because the graph consists of distinct points, the function is discrete. Use the vertical line test. No vertical line can be drawn that contains more than one of the data points. Therefore, the relation is a function. Real-World Example 4

  16. Discrete and Continuous Functions Answer: Yes, this relation is a function, and it is discrete. Real-World Example 4

  17. Choose the Correct Model A commuter train ticket costs $7.25. The cost of taking the train x times can be described by the function y = 7.25x, where y is the total cost in dollars. Determine whether the function is correctly modeled by a discrete or continuous function. Explain your reasoning. When deciding whether a real-world situation is modeled by a discrete or continuous function, consider whether an interval of all real numbers makes sense as part of the domain. Real-World Example 4

  18. Example 5 Choose the Correct Model If a person buys 2 train tickets, it will cost $14.50. Three tickets will cost $21.75, 4 tickets will cost $29.00, and so on. You cannot purchase 1.5 tickets or 2.25 tickets to ride the train. Since the domain consists only of whole number, this is correctly modeled by a discrete function and the graph will consist of the set of unconnected points (1, 7.25), (2, 14.50), (3, 21.75), (4, 29.00), and so on. Because the graph consists of distinct points, the function is discrete.

  19. Example 5 Choose the Correct Model Use the vertical line test. No vertical line can be drawn that contains more than one of the data points. Therefore, the relation is a function. Answer: Discrete; you cannot purchase a fractional of a ticket, so the domain is the set of whole numbers.

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