Domain and Range

Calculus Readiness · Axiom Academy

Domain is the set of inputs a function is allowed to take; range is the set of outputs it can produce — and a graph lets you read both as shadows. Watch a point ride along the graph of y = f(x) . Drop a line straight down from it and you mark an input on the x -axis. Send a line straight across and you mark an output on the y -axis. Let the point sweep the whole curve and those two marks smear into solid bars: the domain is the shadow on the x -axis, the range is the shadow on the y -axis. Domain — every input that is allowed Range — every output that actually occurs Handed a bare formula with no context, the domain is "all real numbers that don't break the formula." Only three things actually break one — everything else (polynomials, sines, cosines, exponentials) is fine for every real number. Division by zero. A denominator can never equal 0 , so exclude any input that zeroes it. Even roots of negatives. Under a square root (or any even root) the expression must be . Logs of non-positives. The argument of a logarithm must be > 0 . The animation shows the first killer in action. The graph of races off to at x = 1 — the denominator is zero there, so x = 1 is punched out of the domain. The shadow on the x -axis is solid everywhere except that one missing point. Set the denominator to zero: . That input is illegal. The domain is everything else, written as two pieces: . Worked example — two killers at once

This is the written version of the interactive lesson above. See the full Calculus Readiness course.