Determining Domain and Range from a Graph

Key Concepts

• Domain: Set of all possible x-values of a function (along the x-axis).
• Range: Set of all possible y-values of a function (along the y-axis).

Determining Domain

1. Project the graph onto the x-axis: Analyze how the graph behaves horizontally.
   • Example 1:
     • Leftmost Point: As x approaches -3 (open point, not included).
     • Rightmost Point: x equals 2 (closed point, included).
     • Domain: From -3 to 2, not including -3, including 2.
     • Inequality Notation: x > -3 and x ≤ 2.
     • Interval Notation: (-3, 2].
   • Example 2:
     • Leftmost Point: x equals -4 (closed point, included).
     • Graph extends indefinitely to the right: Approaching positive infinity.
     • Domain: x ≥ -4.
     • Interval Notation: -4, ∞).

Determining Range

1. Project the graph onto the y-axis: Analyze how the graph behaves vertically.
   • Example 1:
     • Lowest Point: Approaching -5 (open point, not included).
     • Highest Point: y equals 5 (closed point, included).
     • Range: From -5 to 5, not including -5, including 5.
     • Inequality Notation: y > -5 and y ≤ 5.
     • Interval Notation: (-5, 5].
   • Example 2:
     • Lowest Point: y equals -4 (closed point, included).
     • Graph extends indefinitely upwards: Approaching positive infinity.
     • Range: y ≥ -4.
     • Interval Notation: -4, ∞).

Conclusion

• Analyzing the domain and range involves projecting the graph onto the x and y axes.
• Open and closed points determine whether specific values are included or excluded in the domain and range.
• Use interval notation and inequality notation for precise representation.