Domain and Range Analysis in Set Notation

  • To determine the domain and range of a function, follow these steps:   

  1. Identify the Domain

    • Observe the function plotted on the graph (horizontal xx-axis).

    • Find the leftmost and rightmost points of the graph to establish the minimum and maximum xx-values.

    • Use the endpoints to write the domain in set notation.

      • For example: If the leftmost point is −4-4 and the rightmost point is 22, the domain is written as  {x ∣ −4≤x≤2}\text{ }\{x \text{ }| \text{ }-4 \le x \le 2\}.   

  2. Identify the Range

    • Check the vertical span of the function (vertical yy-axis).

    • Determine the lowest and highest points of the graph to find the minimum and maximum yy-values.

    • Express the range in set notation using these endpoints.

      • For example: If the lowest point is −1-1 and the highest point is 55, the range is expressed as {y ∣ −1≤y≤5}\{y \text{ }| \text{ }-1 \le y \le 5\}.

  3. Understand Set Notation

    • Familiarize yourself with the symbols used in set notation:

      • Curly braces {}\{ \} indicate the boundaries of a set.

      • The vertical bar (∣|) represents “such that” in the context of defined values.

    • Use the form {variable ∣ lowerbound≤variable≤upperbound}\{variable \text{ }| \text{ } lower bound \le variable \le upper bound\} to describe intervals accurately.