Mr. IntegralReading Domain and Range off a Graph
Read domain and range from the horizontal and vertical extent of a graph.
The bigger question: What does a function tell us—and what can it hide?
On this page
Idea
Project a graph onto the horizontal axis to read the domain and onto the vertical axis to read the range. Open circles remove particular points; arrows usually indicate continuation beyond the displayed window.
Method
Use brackets for included endpoints and parentheses for excluded endpoints. A missing point removes an output from the range only if no other point on the graph has that same height.
Worked example
Consider the line segment for . Its horizontal projection is . Since the line increases, its vertical projection is . If a graph has a hole at but also passes through , the output still belongs to its range.
Common mistake
Do not interpret the edge of a plotted window as a mathematical endpoint. Look for an explicit domain, endpoint marker or continuation arrow.
Check your understanding
For restricted to , find domain and range.
Show answer
Domain ; range . The minimum occurs at an interior point.
Explore
Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.
Try it yourself.
Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.
Hint 1 · Find a starting point
Read output values vertically, not input values horizontally.
Hint 2 · Take the next step
Every point on this segment has the same height.
Show the reasoning
Answer: {2}
Only y=2 is attained; [−1,3) is the domain.
Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.
Pause before the next idea.
Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.