Radical functions
y = √f(x) exists only where f(x) ≥ 0; it keeps the points where f(x) is 0 or 1, lifts values between them and lowers values above 1.
A worked example
Solve √(x + 7) = x + 1. Squaring: x + 7 = x² + 2x + 1, so x² + x − 6 = 0 and x = 2 or x = −3. Check: at x = 2 both sides are 3. At x = −3 the left side is √4, which is 2, but the right side is −2, so −3 is extraneous. The solution is x = 2 only.
The mistake people actually make
Squaring both sides can add a root that does not work, an extraneous root. Check every answer in the original equation.