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.

This quiz covers the essential concepts of relations and functions, focusing on mapping diagrams and ordered pairs. You'll explore how each input relates to its corresponding output and determine the range of given functions. Test your understanding of these foundational mathematical principles.