Question

The square root of a negative number, such as StartRoot negative 144 EndRoot, is undefined. Explain why the square root of –x, StartRoot negative x EndRoot, is not necessarily undefined and what this means about the domain and range of f(x) = StartRoot negative x EndRoot.

Answer 1

Saying "the square root of a negative number is undefined" means: for real-valued square root, sqrt(A) is only defined when A ≥ 0. For a constant like −144, the radicand is negative so sqrt(−144) is not a real number.

But sqrt(−x) is an expression in the variable x; whether it is defined depends on the value of x, because the radicand is −x, not necessarily a negative constant. The condition for real-valued sqrt(−x) is
−x ≥ 0 ⇔ x ≤ 0.
So sqrt(−x) is defined for all x ≤ 0. For example, when x = −144, −x = 144 and sqrt(−x) = 12.

From this you get the domain and range of f(x) = sqrt(−x) (as a real-valued function):
- Domain: {x : x ≤ 0} = (−∞, 0].
- Range: {y : y ≥ 0} = [0, ∞).

You can also see this by letting y = sqrt(−x). Then y ≥ 0 and y^2 = −x, so x = −y^2 (which is ≤ 0). The graph is the top half of the parabola x = −y^2. (If you allowed complex values, sqrt of a negative constant would be defined in the complex numbers, but here we are working in the real numbers.)