Domain and range
Separate allowed inputs from possible outputs.
The idea
Domain is the set of allowed -values. Range is the set of resulting -values. Read domain across the graph and range up and down.
Check the restrictions
- A real square root needs its radicand to be nonnegative.
- A denominator cannot equal zero.
- Linear, quadratic, cubic, cube-root, and exponential parent functions allow all real inputs.
Learn with an example
Tutor
Work through one example at a time. Turn narration on to listen, and replay whenever you need.
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Example 1
For , require . The domain is and the range is .
Notice and plan
Pause and identify the mathematical feature or graph clue before following the calculation.
Work it through
Take the square root of the quantity x minus three, then add two outside the root. Require x minus three to be greater than or equal to zero. Adding three to both sides gives x greater than or equal to three. The filled endpoint shows that input three is included. A square root is nonnegative. Adding two makes the smallest output two. So the range is y greater than or equal to two. The graph starts with horizontal coordinate three and vertical coordinate two. It then continues upward. Both the domain endpoint and the range endpoint are included.
Check and understand
Substitute x equal to three: the square root is zero and the output is two. Smaller inputs are not allowed, and larger inputs give outputs above two. This checks the domain and range separately.
- Transformed
- Reference lines
Read the narration · spoken math transcript
Example 1. Pause and identify the mathematical feature or graph clue before following the calculation. Take the square root of the quantity x minus three, then add two outside the root. Require x minus three to be greater than or equal to zero. Adding three to both sides gives x greater than or equal to three. The filled endpoint shows that input three is included. A square root is nonnegative. Adding two makes the smallest output two. So the range is y greater than or equal to two. The graph starts with horizontal coordinate three and vertical coordinate two. It then continues upward. Both the domain endpoint and the range endpoint are included. Substitute x equal to three: the square root is zero and the output is two. Smaller inputs are not allowed, and larger inputs give outputs above two. This checks the domain and range separately.
Common mistake
Include an endpoint when the function reaches it. An asymptote is a boundary the relevant branch approaches, not an included endpoint.
Check your understanding
Explain why the rule applies, then check your result in the original problem.