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.
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.