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.
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Answer 1

When we say √(−144) is undefined (in the real numbers) we mean the principal square-root function √t is only defined for nonnegative inputs t, and −144 is negative so there is no real number whose square is −144. But √(−x) is an expression whose radicand depends on x, so it is defined whenever the input to the square root, −x, is nonnegative. That condition is −x ≥ 0 ⇔ x ≤ 0, so the domain of f(x)=√(−x) (over the reals) is all x ≤ 0, and the function values are nonnegative (the square root always returns nonnegative outputs), so the range is [0,∞). In particular f(0)=0 and as x→−∞, f(x)=√(−x)→∞. (If one works in the complex numbers, √(−144) can be given a complex value, but in the real-valued context above it is undefined.)