Domain and Range Notes

Domain and Range: Basics

  • Domain vs Range
    • Domain: the set of all inputs in a relationship
    • Range: the set of all outputs in a relationship
    • Inputs correspond to x-values (horizontal axis on a graph)
    • Outputs correspond to y-values (vertical axis on a graph)
  • Practical example: States and capitals
    • Input (domain): a state, e.g., Nebraska
    • Output (range): its capital, e.g., Lincoln
    • Domain example (states): Alabama, Alaska, Arizona, and so on
    • Range example (capitals): Montgomery, Juneau, Phoenix, and so on
  • Two common ways to represent domain and range
    • Inequality notation
    • Interval notation
  • How to read a graph for domain and range
    • Look at the x-values that appear on the graph for domain
    • Look at the y-values that appear on the graph for range
    • Endpoints and inclusion depend on circle type on endpoints
  • Endpoints: closed vs open circles
    • Closed circle = endpoint is included in the domain/range
    • Open circle = endpoint is not included in the domain/range
  • Example 1: Graph with closed endpoints (endpoints included)
    • Visual cue: endpoints are closed circles
    • Domain (x-values): from 0 to 5 inclusive
    • Range (y-values): from -4 to 5 inclusive
    • Inequality notation
    • Domain: 0≤x≤50 \le x \le 5
    • Range: −4≤y≤5-4 \le y \le 5
    • Interval notation
    • Domain: [0,5][0,5]
    • Range: [−4,5][-4,5]
  • Example 2: Graph with open endpoints on domain, closed endpoint on the upper range
    • Visual cue: endpoints on x-axis are open circles; lower end of range is open, upper end is closed
    • Domain (x-values): from 0 to 5 but not including 0 and 5
    • Range (y-values): from -4 to 5, with -4 not included and 5 included
    • Inequality notation
    • Domain: 0 < x < 5
    • Range: -4 < y \le 5
    • Interval notation
    • Domain: (0,5)(0,5)
    • Range: (−4,5](-4,5]
  • Quick visual interpretation of the two examples
    • Closed endpoints (Example 1): use brackets [ ] in interval notation
    • Open endpoints (Example 2): use parentheses ( ) in interval notation
    • In both cases, the statements describe all possible inputs (domain) and all possible outputs (range) for the graph
  • Summary of key points
    • Domain = set of all inputs (x-values)
    • Range = set of all outputs (y-values)
    • Notation options: inequality notation or interval notation
    • In interval notation, [a,b] means both endpoints included; (a,b) means endpoints not included
    • A closed circle on a graph endpoint indicates inclusion; an open circle indicates exclusion
  • Important conventions and tips
    • Always identify which axis is which: domain corresponds to the x-axis, range to the y-axis
    • Translating between notations:
    • If the graph shows endpoints included: domain = [a,b][a,b], range = [c,d][c,d] and inequalities a≤x≤b,c≤y≤da \le x \le b, c \le y \le d
    • If endpoints excluded: domain = (a,b)(a,b), range = (c,d)(c,d) and inequalities a < x < b, c < y < d
  • Connections to broader topics
    • Domain and range are foundational for understanding relations and functions; for a function, each x in the domain maps to a unique y in the range
    • These concepts also underpin how you interpret real-world data on graphs and how endpoints affect inclusion in solutions
  • Quick reference cheat sheet
    • Domain: x-values on the graph (horizontal axis)
    • Range: y-values on the graph (vertical axis)
    • Notation choices: inequality vs interval
    • Graph endpoint meanings: closed circle = value included; open circle = value not included
    • Example templates:
    • Closed endpoints: Domain [a,b][a,b], Range [c,d][c,d]; Inequality a≤x≤b, c≤y≤da \le x \le b, \ c \le y \le d
    • Open endpoints: Domain (a,b)(a,b), Range (c,d)(c,d); Inequality a < x < b, \ c < y < d

Domain and Range: Additional Examples and Clarifications

  • Reminder: The domain is the set of all inputs; the range is the set of all outputs
  • Both interval notation and inequality notation convey the same information
  • In interval notation, use brackets [] to include endpoints and parentheses () to exclude endpoints
  • Practice takeaway: when you see a graph, immediately identify:
    • The leftmost to rightmost x-values that appear -> domain
    • The bottommost to topmost y-values that appear -> range
  • End of notes on this topic