Definition: The domain of a function is the complete set of input values, usually x-values, for which the function is defined and produces an output.
What is the domain?
Think of a function as a machine. The domain lists every input the machine accepts. For a graph, read it left to right: which x-values have a point above or below them?
What limits a domain?
Two restrictions appear most often in Algebra 1. You cannot divide by zero, and you cannot take an even root of a negative number. Real situations also limit inputs: the number of buses cannot be negative or fractional.
Example. For f(x) = 7 / (x – 9), the denominator is zero when x = 9, so the domain is all real numbers except 9. In interval notation that is (-∞, 9) together with (9, ∞). For g(x) = the square root of (x + 4), we need x + 4 to be at least 0, so the domain is x ≥ -4.
What mistake should you avoid?
Do not confuse domain with range. Domain is about inputs, range about outputs. Also, a word problem may restrict the domain even when the formula would accept more numbers, so always read the context.
Where to learn it
- Understanding Functions and Function Notation – lesson
- Key Features of Graphs and Rates of Change – lesson