Domain and range of real functions

When a real function is given only by a formula, its domain is taken to be the largest set of real numbers for which the formula gives a real value.

Two things shrink the domain: a denominator cannot be 0, and an even root such as √ needs a non-negative quantity under it.

For f(x) = (x² + 2x + 1)/(x² − 8x + 12), the denominator is (x − 2)(x − 6), so the domain is R − {2, 6}.

For f(x) = √(x − 3), we need x − 3 ≥ 0, so the domain is [3, ∞); the range is [0, ∞), because a square root is never negative and x = 3 + k² gives the value k for any k ≥ 0.

For f(x) = √(25 − x²), we need x² ≤ 25, so the domain is [−5, 5]; the value is largest (5) at x = 0 and smallest (0) at x = ±5, so the range is [0, 5].

The range is found by asking which outputs are actually reached. For f(x) = x² + 4 on R the range is [4, ∞); for f(x) = 5 − 2x with x > 1 the range is (−∞, 3), since x > 1 gives 2x > 2 and so 5 − 2x < 3.

A restriction on x changes the range: x² on R has range [0, ∞), but x² on [−1, 2] has range [0, 4].