Domain and range describe the possible inputs and outputs of a relation or function. The domain is the set of x-values that are allowed, and the range is the set of y-values that can actually occur. These ideas matter because they tell you where a formula or graph makes sense.
They also help you interpret real situations, such as time, height, cost, and distance.
Understanding Math: Domain and Range
A useful way to think about a function is as a rule that matches each accepted value with one result. The rule may be shown by a table, a graph, a diagram, or an equation. In a mapping diagram, start with the values on the left.
Follow every arrow to see which results are reached on the right. A value can have several different inputs leading to it. That is allowed.
What is not allowed for a function is one input pointing to two different outputs. This distinction helps when you decide whether a relation is really a function.
Graphs give a quick visual method. To find the allowed horizontal positions, imagine projecting every point of the graph straight down onto the horizontal axis. The covered parts form the input set.
Then project the graph sideways onto the vertical axis to find the output set. Open circles need careful attention. They show a point that is missing, even if the curve comes very close to it.
Arrows usually mean that the graph continues forever in that direction. A vertical line test checks whether the graph represents a function, but it does not by itself find every possible output.
Equations can hide restrictions that are not obvious from their shape. Division creates a problem whenever a bottom expression becomes zero. Even roots require the quantity inside the root to be zero or positive.
Logarithms require their input to be positive. These rules come from the meaning of the operations, not from a teacher-made convention.
For example, no real number multiplied by itself gives a negative result, so an even root of a negative number has no real output. When several operations appear in one formula, check every restriction and combine them.
Finding outputs often takes more work than finding inputs. For a line that is not horizontal, every vertical value can occur. A parabola changes direction at its vertex, so that turning point sets the lowest or highest possible output.
Absolute value graphs have a similar boundary at their corner. For more complicated graphs, use important features such as endpoints, peaks, valleys, holes, and horizontal asymptotes.
An asymptote is a value that a graph may approach forever without reaching. It is not automatically excluded, since some graphs can reach that same value elsewhere.
Real situations add limits beyond the formula. A model for the height of a thrown ball may produce values before it was launched or after it hit the ground. Those values can be mathematically valid but physically irrelevant.
Time is often restricted to zero or greater. The number of people, books, or items must usually be whole numbers. Units matter too.
A cost model may allow decimal outputs, while a count does not. State whether you are describing the full mathematical rule or the realistic situation, since their allowed values may differ.
Key Facts
- Domain = all possible input values, usually the x-values.
- Range = all possible output values, usually the y-values.
- For a fraction, the denominator cannot be zero: if f(x) = 1/(x - 3), then x ≠ 3.
- For an even root, the radicand must be nonnegative: if f(x) = sqrt(x + 4), then x + 4 ≥ 0.
- Interval notation uses brackets for included endpoints and parentheses for excluded endpoints, such as [2, 5) meaning 2 ≤ x < 5.
- For f(x) = ax^2 + bx + c, the vertex helps find the range because a parabola has a minimum if a > 0 and a maximum if a < 0.
Vocabulary
- Domain
- The domain is the set of all input values for which a relation or function is defined.
- Range
- The range is the set of all output values produced by a relation or function.
- Interval notation
- Interval notation is a compact way to write sets of real numbers using brackets, parentheses, and infinity symbols.
- Restriction
- A restriction is a value or condition that must be excluded because it makes an expression undefined or impossible in the real numbers.
- Endpoint
- An endpoint is a boundary value of an interval that may be included or excluded depending on the situation.
Common Mistakes to Avoid
- Confusing domain with range. Domain refers to x-values or inputs, while range refers to y-values or outputs.
- Including values that make a denominator zero. Division by zero is undefined, so those x-values must be excluded from the domain.
- Forgetting radical restrictions for even roots. In real-number functions, expressions under square roots and other even roots must be greater than or equal to zero.
- Using brackets with infinity. Infinity is not a number that can be included, so interval notation always uses parentheses with ∞ and -∞.
Practice Questions
- 1 Find the domain of f(x) = (2x + 1)/(x - 5). Write your answer in interval notation.
- 2 Find the domain and range of f(x) = sqrt(x - 2) + 3. Write both answers in interval notation.
- 3 A graph starts at a filled point at (-4, 1), rises to a highest point at (2, 6), and ends at an open point at (7, -1). Explain the domain and range, including which endpoints are included.