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All courses Algebra I · F-IF.5 23 of 76
Relate a function's domain to its graph and to the real-world quantities it describes.

Find the domain of a continuous graph

Problem
Use the continuous graph below to identify its domain.
Coordinate graph showing a closed horizontal segment from x equals negative 2 to x equals 5. Open full size
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Hint

Track the x-values that actually appear on the graph.

The domain is the set of all input x-values used by the graph. For continuous graphs, that is usually an interval. For discrete graphs, it is a set of listed values.

Solution walkthrough

01

Read horizontal extent

\[\text{graph}~\text{segment}~\text{runs}~\text{from}~x=-2~\text{to}~x=5\]

Domain uses the horizontal coordinates covered by the inspected graph, whose segment begins at -2 and ends at 5.

02

Check continuity between endpoints

\[\text{every}~\text{real}~x~\text{between}~-2~\text{and}~5~\text{appears}\]

The unbroken segment includes all intermediate inputs, so the domain is a continuous interval rather than a two-value set.

03

Check endpoint inclusion

\[\text{filled}~\text{endpoint}~\text{at}~-2;~\text{filled}~\text{endpoint}~\text{at}~5~->~\text{include}~\text{both}\]

Both inspected endpoint markers are closed, so interval notation uses a bracket at each end.

04

State the domain

\[[-2,5]\]

This interval contains exactly every x-value used by the closed continuous segment.

+

Another way

  1. Project the graph vertically onto the x-axis; the shadow covers the closed interval from -2 through 5.

!

Common mistake

Do not list only the endpoints or use parentheses. The segment includes every input between them, and both endpoint dots are filled.

Coordinate graph showing a closed horizontal segment from x equals negative 2 to x equals 5, showing the domain is negative 2 to 5 inclusive.
[-2,5]

Solution walkthrough video