Definition: The range of a function is the set of all output values, usually y-values, that the function actually produces from the inputs in its domain.
How is range different from domain?
Domain lists the inputs. Range lists the outputs that really occur. On a graph, read the range from bottom to top: which y-values are touched by the curve?
How do you find it?
For a table or a set of ordered pairs, list the second numbers without repeats. For a graph, find the lowest and highest points, or note that the graph continues forever. For a formula, think about which outputs are possible.
Example. Take f(x) = x2 – 3. Squares are never negative, so x2 is at least 0, and f(x) is at least -3. The range is y ≥ -3. For the pairs (1, 6), (2, 6), (3, 9), the range is {6, 9}, because 6 is listed only once.
What do students get wrong?
They list repeated outputs twice, or they report the domain instead. A good check is to ask: does some input really give this output? If not, the number is not in the range.
Where to learn it
- Understanding Functions and Function Notation – lesson
- Key Features of Graphs and Rates of Change – lesson