Key takeaway: The domain and range of a function are the complete sets of allowed inputs (x-values) and resulting outputs (y-values), respectively.
Grade 9 – Algebra 1F-IF.A.1Functions and Graphs
Definition of Domain and Range
The domain and range describe the two sides of a function. The domain lists every input the function accepts. The range lists every output that the function actually produces.
On a graph, read the domain left to right along the x-axis and the range bottom to top along the y-axis. Some inputs are banned because they break math rules: you cannot divide by zero, and you cannot take the square root of a negative number in the real numbers. Real situations also limit inputs, since a number of tickets cannot be negative.
Formula and notation
Write both sets with inequalities or interval notation:
\[\text{domain: } x \ge 3 \quad\text{or}\quad [3, \infty)\]
Parentheses exclude an endpoint, brackets include it, and \(\infty\) always gets a parenthesis. A list such as \(\{1, 4, 9\}\) works when the values are separate.
Key facts
- Domain is the horizontal spread of the graph; range is the vertical spread.
- Division by zero and even roots of negatives remove values from the domain.
- A function can have a limited range even when its domain is all real numbers.
- Word problems add practical limits, such as time that cannot be negative.
Domain and Range example
Find the domain and range of \(f(x) = \sqrt{x – 3} + 2\).
-
The expression under the root must not be negative: \(x – 3 \ge 0\), so \(x \ge 3\).
-
At \(x = 3\), \(f(3) = 0 + 2 = 2\). A square root is never below 0, so \(f(x)\) is never below 2.
-
As \(x\) grows, \(f(x)\) grows without bound, so every number from 2 upward is reached.
Domain: \(x \ge 3\). Range: \(y \ge 2\).
Common mistake
Watch out
Samir states that the range of \(f(x) = \sqrt{x – 3} + 2\) is \(y \ge 0\) because “square roots start at 0”.
Why it is wrong
The root starts at 0, but the function adds 2 afterward, shifting every output up by 2. The smallest output is \(f(3) = 2\), so the range is \(y \ge 2\). Trace the lowest point of the graph instead of guessing from the root alone.
Related terms and lessons
- Function
- Function Notation
- Vertical Line Test
- Independent Variable
- Dependent Variable
- Absolute Value Function
- Algebra 1 · Functions, Domain, Range and Function NotationLessonWorksheetQuiz
- Precalculus · Functions: Domain, Range, Composition and InversesLessonWorksheetQuiz
Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs