Domain and range

f(x) = √(x + 2) has no real output at x = −3, because −3 + 2 = −1 and a negative number has no real square root. Its domain starts at −2.

What comes out when −3 goes into √(x + 2)?

It is tempting to think that a formula accepts any number. Try one. Put x = −3 into f(x) = √(x + 2). Under the root you get −3 + 2 = −1, and no real number squared gives −1. Nothing comes out. So the formula itself rules some inputs out, and the inputs it does accept form a set with its own name.

Domain is what goes in, range is what comes out

The domain of a function is the set of inputs it accepts, and the range is the set of outputs it can produce. For a list of pairs you read them off. OpenStax gives the function {(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)}: its domain is {2, 3, 4, 5, 6} and its range is {10, 20, 30, 40}. The value 10 appears twice as an output but is listed once in the range. For a formula, the domain is usually every real number that does not break a rule of arithmetic.

Two rules decide most domains

For real numbers there are two rules to watch. You cannot divide by 0, and you cannot take an even root, such as a square root, of a negative number. OpenStax turns this into a short check. If the formula has no denominator and no even root, the domain is probably all real numbers. If it has a denominator, remove every x that makes the denominator 0. If it has an even root, keep only the x that make the expression under the root zero or positive. Real situations add their own limits: a number of tickets cannot be negative.

Examples make it concrete. For f(x) = 1/(x − 1), the denominator is 0 at x = 1, so the domain is every real number except 1. For f(x) = √(x + 2), we need x + 2 ≥ 0, so x ≥ −2. For f(x) = √(6 − x), we need 6 − x ≥ 0, so x ≤ 6. For f(x) = x² − 4 there is nothing to avoid and the domain is all real numbers; its range, though, is not everything, because x² is never negative, so x² − 4 is never below −4.

Brackets keep an endpoint, parentheses drop it

Domains and ranges are often whole stretches of the number line, and interval notation writes them compactly. The smaller endpoint comes first, then a comma, then the larger. A square bracket [ or ] means the endpoint is included; a parenthesis ( or ) means it is left out. Infinity is not a number you can reach, so it always gets a parenthesis. OpenStax's example: if you have $100 to spend, the amount you can spend is more than 0 and at most 100, written (0, 100]. The domain of √(x + 2) is [−2, ∞), and the domain of 1/(x − 1) is (−∞, 1) ∪ (1, ∞), where ∪ joins two pieces.

Domain and range as shadows on the two axes

On a graph, the domain is the shadow the curve casts on the x-axis, and the range is its shadow on the y-axis. Squash the graph flat onto each axis and see what is covered. In the lab the green bar on the x-axis is the domain and the orange bar on the y-axis is the range. A filled dot at an end means that endpoint belongs to the set; an open dot means it does not, which is the open circle you see at x = 1 for 1/(x − 1). Move the x slider into a forbidden region and the panel tells you which rule the input breaks.