Domain and Range
Domain is all valid inputs; range is all outputs.
About this topic
The domain of a function is the set of all allowed inputs, and the range is the set of all possible outputs. For f(x) = √x, the domain is x ≥ 0 because you cannot take the square root of a negative, and the range is y ≥ 0.
Typical restrictions are division by zero and even roots of negative numbers. For f(x) = 1/(x − 3), the domain is all real numbers except 3, since that would make the denominator zero. You can read the domain from a graph as the left-to-right spread, and the range as the bottom-to-top spread. Interval notation, such as [0, ∞), is a compact way to write these sets.
Real situations add practical limits. If h(t) gives the height of a thrown ball, time cannot be negative and the ball cannot go below the ground, so the domain and range are restricted by the context. A store cannot sell a negative number of items.
A frequent error is to confuse the two sets. Ask yourself what values x may take for domain and what values y reaches for range. For a parabola with vertex at (0, 2) opening downward, the range is y ≤ 2, found from the highest point.
Key ideas
- Avoid division by zero
- No negatives under square roots
- Read range from the graph's heights
Quick check
Use cases: where Domain and Range helps
- Homework: review the key idea behind Domain and Range before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Domain and Range without notes.
- Visual learning: play with the interactive lab to see how Domain and Range changes when you adjust its inputs.
- Teaching a friend: explain Domain and Range in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Domain and Range?
Domain is all valid inputs; range is all outputs.
What will I learn in Domain and Range?
Avoid division by zero. No negatives under square roots. Read range from the graph's heights.
Is Domain and Range part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Domain and Range lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Domain and Range?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.