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2.1 Functions and Their Graphs

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2.1 Functions and Their Graphs

  1. 1. 2.1 Functions and Their Graphs
  2. 2. What is a Relation?• A relation is a mapping, or pairing, of input values with output values.  “Mapping Diagram”• The set of input values is the domain.• The set of output values is the range.• What are the domain and range of this relation?
  3. 3. How Can We Write a Relation?• A relation can be written in the form of a table:• A relation can also be written as a set of ordered pairs:
  4. 4. How Do We Write a Relation with Numbers? • Set of ordered pairs with form (x, y). • The x-coordinate is the input and the y- coordinate is the output. • Example: { (0, 1) , (5, 2) , (-3, 9) } • { } is the symbol for a “set” • What is the domain and range of this relation?
  5. 5. How Do We Graph a Relation?• To graph a relation, plot each of its ordered pairs on a coordinate plane.• Graph the relation: { (0, 1) , (5, 2) , (-3, 9) } Remember: The x comes first – moves right or left. The y comes second – moves up or down. Positive means to the right or up. Negative means to the left or down.
  6. 6. Your Turn!• Graph the relation and identify the domain and range. { (-1,2), (2, 5), (1, 3), (8, 2) }
  7. 7. What is a Function?• A function is a special type of relation that has exactly one output for each input.• If any input maps to more than one output, then it is not a function.• Is this a function? Why or why not?
  8. 8. Which of These Relations Are Functions? • • { (3,4), (4,5), (6,7), (3,9) } • X 5 7 9 2 6 y 1 6 2 8 4
  9. 9. Using the Vertical Line Test• A relation is a function if and only if no vertical line crosses the graph at more than one point.• This is not a function because the vertical line crosses two points.
  10. 10. Your Turn!• Write the domain and range.• Is this a function? { (2,4) (3,6) (4,4) (5, 10) } stop
  11. 11. What is a Solution of an Equation?• Many functions can be written as an equation, such as y = 2x – 7.• A solution of an equation is an ordered pair (x, y) that makes the equation true.• Example: Is (2, -3) a solution of y = 2x – 7 ?
  12. 12. What are Independent and DependentVariables?• The input is called the independent variable. ▫ Usually the x• The output is called the dependent variable. ▫ Usually the y• Helpful Hints: ▫ Input and Independent both start with “in” ▫ The Dependent variable depends on the value of the input
  13. 13. What Does the Graph of an Equation Mean? • The graph of a two variable equation is the collection of all of its solutions. • Each point on the graph is an ordered pair (x, y) that makes the equation true. • Example: This is the graph of the equation y = x + 2
  14. 14. How Do We Graph Equations?• Step 1: Construct a table of values.• Step 2: Graph enough solutions to notice a pattern.• Step 3: Connect the points with a line or curve.
  15. 15. Example:• Graph the equation y = x + 1
  16. 16. Your Turn!• Graph the equation y = x – 2
  17. 17. What is Function Notation?• Function notation is another way to write an equation.• We can name the function “f” and replace the y with f(x).• f(x) is read “f of x” and means “the value of f at x.” ▫ Be Careful! It does not mean “f times x”• Not always named “f”, they sometimes use other letters like g or h.
  18. 18. What is a Linear Function?• A linear function is any function that can be written in the form f(x) = mx + b• Its graph will always be a straight line.• Are these functions linear? ▫ f(x) = x2 + 3x + 5 ▫ g(x) = 2x + 6
  19. 19. How Do We Evaluate Functions?• Plug-in the given value for x and find f(x).• Example: Evaluate the functions when x = -2. ▫ f(x) = x2 + 3x + 5 ▫ g(x) = 2x + 6
  20. 20. Your Turn!• Decide if the function is linear. Then evaluate the function when x = 3. g(x) = -3x + 4 Stop?
  21. 21. How Do We Find the Domain and Range?• The domain is all of the input values that make sense. ▫ Sometimes “all real numbers” ▫ For real-life problems may be limited• The range is the set of all outputs.
  22. 22. Example:• In Oak Park, houses will be from 1450 to 2100 square feet. The cost C of building is $75 per square foot and can be modeled by C = 75f, where f is the number of square feet. Give the domain and range of C(f).