Introduction to Functions: Part Two

Graphing Functions, Domain, and Range

Graphing a Function

• Example: Function ( f(x) = 4x - 2 )
  • Can be rewritten as ( y = 4x - 2 ) (( f(x) = y ))
  • Options for graphing:
    • Slope-intercept form
    • T-table
  • T-Table Example:
    • Select x-values: 0, 1, 2, -1
    • Calculate y-values:
      • ( 4 \times 0 - 2 = -2 )
      • ( 4 \times 1 - 2 = 2 )
      • ( 4 \times 2 - 2 = 6 )
      • ( 4 \times -1 - 2 = -6 )
    • Plot points: (0, -2), (1, 2), (2, 6), (-1, -6)
  • Function notation:
    • ( f(0) = -2 )
    • ( f(1) = 2 )
    • ( f(2) = 6 )

Domain and Range

• Domain

  • Set of all possible x-values (inputs)
  • For linear functions like ( f(x) = 4x - 2 ), domain is all real numbers.
  • Graphically, moves left and right indefinitely.

• Range

  • Set of all possible y-values (outputs)
  • For linear functions, range is all real numbers.
  • Graphically, moves up and down indefinitely.

Quadratic Function Example

• Function: ( f(x) = x^2 )

  • Rewrites as ( y = x^2 )
  • T-Table Example:
    • Select x-values: -2, -1, 0, 1, 2
    • Calculate y-values by squaring x:
      • ( -2^2 = 4 )
      • ( -1^2 = 1 )
      • ( 0^2 = 0 )
      • ( 1^2 = 1 )
      • ( 2^2 = 4 )
    • Plot points: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)
  • Graph forms a U-shape (parabola)

• Domain and Range for Quadratic

  • Domain: All real numbers
  • Range: y-values ( \geq 0 )
    • Lowest point at y = 0, graph extends upwards

Absolute Value Function Example

• Function: ( f(x) = |x| - 3 )

  • Rewrites as ( y = |x| - 3 )
  • T-Table Example:
    • Select x-values: -2, -1, 0, 1, 2
    • Calculate y-values:
      • ( |-2| - 3 = -1 )
      • ( |-1| - 3 = -2 )
      • ( |0| - 3 = -3 )
      • ( |1| - 3 = -2 )
      • ( |2| - 3 = -1 )
    • Plot points: (-2, -1), (-1, -2), (0, -3), (1, -2), (2, -1)
  • Graph forms a V-shape

• Domain and Range for Absolute Value

  • Domain: All real numbers
  • Range: y-values ( \geq -3 )
    • Lowest point at y = -3, graph extends upwards

Verifying Graphs with a Graphing Calculator

• Function notation: ( f(x) = y ), equations are solved for y
• Steps for verification:
  • Enter equation into calculator
  • Use graphing function to visualize
  • Use T-Table function in calculator to verify points

Conclusion

• Understanding graphs in function notation and the concepts of domain and range are fundamental.
• Practice with various functions to solidify understanding.
• Use graphing calculators as a tool for verification and exploration.