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2.1 Functions and Their Graphs
 

2.1 Functions and Their Graphs

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    2.1 Functions and Their Graphs 2.1 Functions and Their Graphs Presentation Transcript

    • 2.1 Functions and Their Graphs
    • 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?
    • 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:
    • 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?
    • 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.
    • Your Turn!• Graph the relation and identify the domain and range. { (-1,2), (2, 5), (1, 3), (8, 2) }
    • 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?
    • 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
    • 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.
    • Your Turn!• Write the domain and range.• Is this a function? { (2,4) (3,6) (4,4) (5, 10) } stop
    • 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 ?
    • 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
    • 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
    • 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.
    • Example:• Graph the equation y = x + 1
    • Your Turn!• Graph the equation y = x – 2
    • 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.
    • 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
    • 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
    • Your Turn!• Decide if the function is linear. Then evaluate the function when x = 3. g(x) = -3x + 4 Stop?
    • 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.
    • 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).