Square root and cube root functions and equations

The square root function f(x) = square root of x has domain x >= 0 and range y >= 0, starting at (0, 0) and curving upward ever more slowly; it is the inverse of y = x^2 restricted to x >= 0. The cube root function f(x) = cube root of x is defined for all real numbers, passing through (-8, -2), (0, 0), (1, 1), and (8, 2), and it is the inverse of y = x^3. Transformations work as for other functions: for either parent, af(x) stretches or reflects vertically, f(bx) stretches, compresses, or reflects horizontally, f(x - c) shifts right by c, and f(x) + d shifts up by d, so y = 2 times the square root of (x - 1), plus 3, starts at (1, 3), and y = the cube root of (x + 2), minus 1, is centered at (-2, -1). With technology, a square root model can also be fit to data that grow quickly at first and then level off more and more slowly. To solve a square root equation, isolate the radical, square both sides, and then check every answer, because squaring can create extraneous solutions: for the square root of (x + 3) = x - 3, squaring gives x^2 - 7x + 6 = 0, so x = 1 or x = 6, but only x = 6 works (x = 1 gives 2 = -2). To solve a cube root equation, cube both sides: the cube root of (x - 2) = 3 gives x - 2 = 27, so x = 29.

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