Domain and Range for Secondary Mathematics Students

Domain is the permitted input set

For y = 2x + 3 considered as a real-valued algebraic function, every real x can be used, so the domain may be all real numbers. But the same formula in a context can have a narrower domain.

If x counts tickets, x may be restricted to nonnegative integers. If x is elapsed time during a ten-minute experiment, a model may use 0 ≤ x ≤ 10. The formula alone does not always tell the complete domain.

Range is the set of possible outputs

For y = x² with real x, the output can never be negative, so the range is y ≥ 0. Every nonnegative real value is possible: if y = a ≥ 0, choosing x = √a gives that output.

If the domain is instead −2 ≤ x ≤ 3, the range remains 0 ≤ y ≤ 9 because x = 0 gives the minimum 0 and x = 3 gives the maximum 9.

Denominators create exclusions

For f(x) = 1/(x − 4), x = 4 is not permitted because the denominator would be zero. The domain is all real numbers except 4.

The output can never be zero because 1 divided by a nonzero real number is never zero. Thus zero is excluded from the range.

Square roots create real-domain conditions

For y = √(x − 2) as a real-valued function, x − 2 must be nonnegative. Therefore x ≥ 2. The principal square root is nonnegative, so y ≥ 0.

A graph can show domain and range

Read domain horizontally: which x-values occur on the graph? Read range vertically: which y-values occur? Endpoints, holes and restrictions matter. A graph drawn only in a limited viewing window should not automatically be assumed to stop at the edge unless the graph or context says so.

Context changes the mathematical object

Suppose h = 20 − 2t models the height of a quantity over a stated interval from t = 0 until h reaches 0. Algebraically the line continues forever, but the model may restrict time to 0 ≤ t ≤ 10 and height to 0 ≤ h ≤ 20.

Using t = 50 would produce h = −80 algebraically, but that output may have no meaning in the stated model. Domain is part of modelling discipline.

Domain restrictions must survive simplification

(x² − 9)/(x − 3) simplifies to x + 3 for x ≠ 3. The simplified expression x + 3 by itself accepts x = 3, but the original function did not. If the simplified formula is being used to represent the original function, retain the exclusion.

Worked examples

A. y = 5x − 1, real x: domain all real numbers, range all real numbers.

B. y = x² + 4, real x: domain all real numbers, range y ≥ 4.

C. y = 1/x, real-valued: domain x ≠ 0, range y ≠ 0.

D. y = √x, real-valued: domain x ≥ 0, range y ≥ 0.

A domain-range routine

1. Identify the proposed input variable.
2. Check algebraic restrictions.
3. Add contextual restrictions.
4. Determine which outputs are actually attainable.
5. Retain restrictions after rewriting or simplifying.

Continue

Build from Input, Output and Dependency and One Relationship, Four Representations, then continue to graph/function testing. Return to the Secondary Mathematics Master Index.