Domain and range describe the input and output values of a function. Students need this cheat sheet because these ideas appear in graphs, tables, equations, word problems, and function transformations. Knowing how to identify domain and range helps you decide which values make sense and which values are impossible.
This is especially important when functions include fractions, square roots, graphs, or real-world limits.
The domain is the set of allowed -values, and the range is the set of possible -values. Common restrictions come from denominators that cannot equal , even roots that cannot have negative radicands, and context-based limits such as time or distance. Graphs show domain from left to right and range from bottom to top.
Interval notation, set notation, and inequalities are common ways to write final answers.
Key Facts
- The domain of a function is the set of all possible input values, usually the allowed -values.
- The range of a function is the set of all possible output values, usually the resulting -values.
- A denominator cannot equal , so for the domain excludes .
- For an even root such as , the radicand must satisfy , so .
- For a linear function with no stated restrictions, the domain is and the range is when .
- For a quadratic function , the range is if and if .
- On a graph, read domain from the farthest left -value to the farthest right -value, and read range from the lowest -value to the highest -value.
- Use brackets when an endpoint is included and parentheses when an endpoint is not included.
Vocabulary
- Domain
- The domain is the set of all input values that a function is allowed to use.
- Range
- The range is the set of all output values that a function can produce.
- Interval Notation
- Interval notation is a compact way to describe a set of numbers using endpoints, brackets, parentheses, and infinity symbols.
- Restriction
- A restriction is a value or condition that is not allowed because it would make the function undefined or impossible in context.
- Undefined
- An expression is undefined when it has no valid mathematical value, such as division by .
- Endpoint
- An endpoint is a boundary value of an interval that may be included or excluded from the set.
Common Mistakes to Avoid
- Confusing domain and range is wrong because domain describes inputs , while range describes outputs .
- Including values that make a denominator is wrong because division by is undefined, so those -values must be excluded.
- Forgetting the radicand restriction in is wrong because an even root requires the expression inside the radical to be nonnegative.
- Using brackets with or is wrong because infinity is not an actual endpoint, so it must always use parentheses.
- Reading graph endpoints incorrectly is wrong because a closed dot means the endpoint is included, while an open dot means it is not included.
Practice Questions
- 1 Find the domain of .
- 2 Find the domain and range of .
- 3 Find the range of .
- 4 A function models the height of a ball after it is thrown and stops being tracked when it hits the ground. Explain why the domain and range should be limited by the real situation, not only by the equation.
Understanding Domain & Range of Functions
A function has one important rule. Each allowed input must produce only one output. Different inputs may still lead to the same output.
This is why a parabola can be a function even though many horizontal lines cross it twice. The vertical line test checks the actual function rule on a graph. A vertical line that hits the graph more than once shows that one input has more than one output.
That relation is not a function. The horizontal line test answers a different question.
It tells whether a function has an inverse that is itself a function. Do not use it to decide whether the original graph is a function.
Finding possible outputs can require more thought than finding allowed inputs. One useful method is to begin with an output value and work backward. Ask which output values would force an impossible step when you solve for the input.
For example, a reciprocal relationship can get extremely close to zero without ever producing zero. That missing output belongs in neither part of the graph. For a quadratic, the turning point creates a highest or lowest output.
Shifting a graph upward changes that boundary. Stretching or reflecting the graph can change which direction the outputs extend. Learning to connect the equation to its parent graph makes these patterns much easier to predict.
Graphs can contain details that change an answer completely. An open circle marks a point that is missing, even if the curve appears to continue around it. A filled circle marks a point that is included.
A single filled point can add one allowed input or one possible output that the nearby curve does not show. Arrows mean the graph continues beyond the viewing window. They are evidence that there is no final endpoint in that direction.
Vertical asymptotes show inputs near a certain value, but not at that value. Horizontal asymptotes need more care.
A graph may approach a horizontal level forever, or it may cross that level somewhere else. Always inspect the whole graph before deciding whether an output is excluded.
Real situations often place limits on a model that the equation alone does not show. A formula for a taxi fare may accept only travel distances of zero or more. A model for a school event may use whole numbers because half a student has no meaning.
Time may start at zero and end when an experiment stops. These are discrete domains, where separate values are used, rather than continuous domains, where every value in an interval is possible. Tables need the same care.
A table may list only measured values, not every value between them. When checking an answer, identify the type of representation, locate exclusions and endpoints, then decide whether the situation requires all real values, an interval, or a list of separate values.