Domain and Range of a Function

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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