functions

• Domain


  The set of first coordinates in an ordered pair
• Range


  The set of second coordinates in an ordered pair
• Finding domain and range from ordered pairs

  1. Identify the set of x-values (domain)

  2. Identify the set of y-values (range)
• Relation


  A set of ordered pairs
• Function


  A relation in which each element of the domain corresponds to exactly one element of the range
• Ordered pair relations


  • Relation F (1, 2), (2, 2), (3, 5), (4, 5) - is a function
  • Relation G (1, 3), (1, 4), (2, 5), (2, 6), (3, 7) - is not a function
  • Relation H (1, 3), (2, 6), (3, 9), (n, 3n) - is a function
• Identifying functions from ordered pairs

  • Each x-value (domain) corresponds to only one y-value (range)
• Mapping diagrams

  • Diagram 1 - is a function
  • Diagram 2 - is not a function
  • Diagram 3 - is a function
• Vertical line test


  • A graph represents a function if and only if each vertical line intersects the graph at most once
• Graphs

  • Graph 1 - is a function
  • Graph 2 - is not a function
  • Graph 3 - is not a function
• Equations

  • y = 2x + 1 - is a function
  • x^2 + y^2 = 1 - is a function
  • x^2 + y - 4 = 0 - is a function
  • y = sqrt(x + 1) - is a function
  • y = 2x + 1 / (x - 1) - is a function
  • x^2 + y^2 = 1 - is not a function
• Vertical line test


  A method to determine if a graph represents a function
• Vertical line test


  • Small alum points
  • Tata man one two three four five six
• A big civilian that is not a function
• Function


  A relationship where each input has exactly one output
• How to identify if a graph is a function or not

  1. Digitized Salus representation

  2. Use equation
• Equations to determine if they represent a function

  • y = 2x + 1
  • x^2 + y^2 = 1
  • x^2 + y - 4 = 0
  • y = x^2 - 2x + 2
  • y = √(x + 1)
  • y = 2x + 1 / (x - 1)
• Linear equation


  A function because the graph is a straight line
• Quadratic equation


  A function because the graph is an upward parabola
• Rational equation


  A function because the graph has no vertical asymptotes
• x^2 + y^2 = 1 is not a function
• How to determine if x^2 + y^2 = 1 is a function

  1. Solve for y in terms of x

  2. Use the square root method
• Functions can be represented using ordered pairs
• How to check if a set of ordered pairs represents a function

  Use the vertical line test
• Mapping diagrams can also represent functions
• Graphs can also represent functions
• Examples of graphs

  • Graph 1: Not a function
  • Graph 2: Function
• Equations that represent functions

  • y = 2x + 1
  • y^2 + x^2 = 9
• y^2 + x^2 = 9 is not a function
• Function

  
  
• A set of ordered pairs where the first element (the input) determines the second element (the output). A rule that takes an input and gives a corresponding output.
• Equation

  
  
• A statement that two expressions are equal, typically involving variables, constants, and mathematical operations.