Domain and Range of a Function

The domain of a function is the complete set of input values (usually x) it can accept. The range is the complete set of output values (usually y) it can produce. Together they tell you exactly what a function can take in and give back, so finding them is a core skill once you can start identifying functions.

Start by assuming the domain is all real numbers, then remove any x-values that break the function. Two rules cover most cases: you cannot divide by zero, and you cannot take the square root of a negative number. For f(x) = 1/(x − 3), the value x = 3 makes the denominator zero, so the domain is every real number except 3. For f(x) = √x, negative inputs are not allowed, so the domain is x ≥ 0.

The range is the set of y-values the function actually reaches. A graph is the quickest guide: read how high and how low the curve goes. The parabola f(x) = x² never dips below zero, so its range is y ≥ 0. Reading the range often depends on how the function is written, which is why comfort with function notation helps.

Notice the pattern: linear functions accept and return all real numbers, while square roots and reciprocals carry restrictions. When you build an inverse function, the domain and range simply swap places — the inputs of the inverse are the outputs of the original.

You can describe domain and range in words, with inequalities, or with interval notation. "All real numbers" is written (−∞, ∞); "y ≥ 0" is written [0, ∞). Square brackets include the endpoint; parentheses exclude it.

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