Mathematics Relations and Functions

Relations and Functions

• A mapping diagram visually represents the relationship between a set of inputs (domain) and outputs (range).
• A set of ordered pairs follows a specific rule, indicating how each input is associated with its output.
• The set of all y-values in a function is called the range.

Key Concepts in Mapping Diagrams

• Different types of mapping diagrams are used to show various functions and relationships.
• The image of a specific input ( x ) is denoted by ( f(x) ), indicating the output value corresponding to that input.

Problem-Solving Techniques

• To find the range ( Y ) for a relation like ( x \to x + 3 ) given the domain ( X = {0, 1, 2, 3} ), substitute each value of ( x ) into the function:
  • For ( x = 0 ), ( Y = 0 + 3 = 3 )
  • For ( x = 1 ), ( Y = 1 + 3 = 4 )
  • For ( x = 2 ), ( Y = 2 + 3 = 5 )
  • For ( x = 3 ), ( Y = 3 + 3 = 6 )
  • Resulting range ( Y = {3, 4, 5, 6} )

Evaluating Functions

• To evaluate the function ( f(x) = 3x^2 + 2x - 1 ) at ( x = 3 ):
  • Substitute ( 3 ) into the function:
  • ( f(3) = 3(3)^2 + 2(3) - 1 = 27 + 6 - 1 = 32 ).

Quadratic Functions

• The quadratic function ( f(x) = 2x^2 ) requires completion of a table for the domain ( -4 \leq x \leq 4 ):
  • Calculate ( f(x) ) for each ( x ) value:
    • ( f(-4) = 32 )
    • ( f(-3) = 18 )
    • ( f(-2) = 8 )
    • ( f(-1) = 2 )
    • ( f(0) = 0 )
    • ( f(1) = 2 )
    • ( f(2) = 8 )
    • ( f(3) = 18 )
    • ( f(4) = 32 )

Graphing Quadratic Functions

• Graph the function using the calculated values in the table to visualize the parabolic shape of quadratic functions, ensuring accuracy by plotting key points along the x-axis from -4 to 4.