Domain and Range Calculator

The domain is the set of valid x-values you can put into a function, and the range is the set of y-values the function can produce. For f(x) = sqrt(x), the domain is x >= 0 because you cannot take a real square root of a negative number, and the range is y >= 0 because the square root symbol returns the non-negative root.

Domain and range come from the algebraic shape of a function, not from a single formula. Picking a family (linear, quadratic, square root, rational, log, and so on) lets the calculator apply the known rule for that shape instead of guessing. It is faster and far more reliable than trying to read a general expression.

Set the denominator equal to zero and exclude any x-value that makes it zero. For 1 / (x - 3), the denominator is zero at x = 3, so the domain is all real numbers except 3, written (-inf, 3) U (3, inf). The range of this particular rational function excludes y = 0 because 1 divided by any non-zero number is non-zero.

A quadratic in vertex form f(x) = a(x - h)^2 + k opens upward when a > 0 and downward when a < 0. The vertex is the lowest point (a > 0) or highest point (a < 0), so the range is bounded below at y = k or bounded above at y = k. The other side runs to infinity.

For f(x) = a * b^x + k with b > 0 and b != 1, the graph approaches y = k as a horizontal asymptote but never touches it. The range is (k, infinity) when a > 0 and (-infinity, k) when a < 0. The domain is always all real numbers.