Domain and range
The domain lists allowed inputs; the range lists outputs actually attained.
Inspect denominators, even roots and any stated restrictions before finding a range. A formula alone may not specify the intended function. A domain restriction can change its range and can determine whether an inverse function exists.
Do not report all real outputs merely because the input is real. For a quadratic, locate the vertex and consider the allowed input interval. For an open endpoint, decide whether its output is attained elsewhere before choosing an open or closed range endpoint.
Worked example
Find the range of f(x) = (x − 2)² + 1 for x ≥ 3.
- For x ≥ 3, x − 2 ≥ 1.
- The smallest square is 1, attained at x = 3.
- The function increases without an upper bound on this domain.
Answer: f(x) ≥ 2
Practise domain and range
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 2.3: Domain and range
- 7M20 · core · A2.3: Domain and range
- 8365 · core · 2.3: Range of a quadratic on a restricted interval
- 7M20 · core · A2.3: Range of a quadratic on a restricted interval
Next practice: Inverse functions, Composite functions.