Algebra
Relations and functions
Domain and range
20 practice questions
2 video lessons
Theory + worked examples
Theory
For a function:
- The domain is the set of all input (\(x\)) values.
- The range is the set of all output (\(y\)) values.
Read a graph left-to-right for the domain and bottom-to-top for the range.
Domain on the \(x\)-axis, range on the \(y\)-axis.
Finding domain and range.
Domain and range:
\[\text{domain}=\{x\},\qquad \text{range}=\{y\}\]
Watch for restrictions like division by zero or square roots.
How to find domain and range
- For a set of pairs, list the inputs and outputs.
- For a graph, read the \(x\)- and \(y\)-extents.
- Note any restricted values.
- Write in set or interval notation.
Example 1 — From a set
Give the domain and range of \(\{(1,2),(2,4),(3,6)\}\).
Solution
Domain is the inputs, range the outputs.
| \(\text{domain}\) | \(=\) | \(\{1,2,3\}\) |
| \(\text{range}\) | \(=\) | \(\{2,4,6\}\) |
Example 2 — Domain of a graph
A line segment runs from \(x=1\) to \(x=6\). Give the domain.
Solution
Read the \(x\)-extent.
| \(\text{domain}\) | \(=\) | \([1,6]\) |
Example 3 — Range of a parabola
Give the range of \(y=x^2\).
Solution
Outputs are never negative.
| \(\text{range}\) | \(=\) | \([0,\infty)\) |
Example 4 — All real domain
What is the domain of \(y=2x+1\)?
Solution
A line accepts any input.
| \(\text{domain}\) | \(=\) | \((-\infty,\infty)\) |
Common pitfalls
Domain is inputs (\(x\)), range is outputs (\(y\)).
A parabola's range is limited by its vertex.
Exclude values that break the function.
Frequently asked questions
What is the domain?
The set of all input \(x\)-values.
What is the range?
The set of all output \(y\)-values.
What is the range of \(y=x^2\)?
\([0,\infty)\).
How do you read the domain from a graph?
Look at the left-to-right extent along the \(x\)-axis.
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