Piecewise and Radical Functions

A phone plan that charges one rate for the first 5 gigabytes and a different rate after that cannot be written as one clean formula, because the rule itself changes depending on how much data you use. That is exactly what a piecewise function is: a function built from separate formulas, each one valid only over its own piece of the domain. To evaluate it at a given input, you first find which piece that input belongs to, using the inequalities attached to each formula, then plug into that formula only. Get the piece wrong and you get a completely wrong output, even if your algebra is perfect.

A common worry is that a piecewise graph with a jump in it cannot be a function, since the two pieces do not meet. But the function rule only cares about outputs, not about whether the picture looks connected. At the exact x-value where the jump happens, one piece is usually drawn with an open circle (not included) and the other with a closed circle (included), so that x-value still gets exactly one output. A jump breaks continuity, not the definition of a function.

Radical functions connect to this same idea of domain restriction. A square root, index 2, cannot take a negative input in the real numbers, so its domain stops wherever the inside expression goes negative. A cube root, index 3, has no such problem — negative numbers have real cube roots — so its domain is all real numbers. Changing the index from even to odd is what removes the restriction entirely.