Radical functions
A function that takes a root of its input may turn some inputs away. Where does its graph begin, how does it bend, and what does it describe?
Where you are 3 inside
What this area is about
A radical function puts its input under a root, as in \displaystyle f(x) = \sqrt{x} or \displaystyle g(x) = \sqrt[3]{x}. With a square root, only inputs that make the expression under the root at least 0 are allowed. For a rule such as \displaystyle f(x) = a\sqrt{x - h} + k the graph is a half curve: it begins at an endpoint and runs off in one direction, flattening as it goes. A cube root accepts every real number, so its graph runs in both directions.
How these fit together
Graphs. Graphing square root functions finds the domain, the endpoint and the range of \displaystyle f(x) = a\sqrt{x - h} + k with a \ne 0, and draws the half curve. Cube root functions and roots as powers graphs the cube root, which accepts every input, and reads x^{1/3} as \displaystyle \sqrt[3]{x}.
Applications. Radical functions in context reads and builds models such as wave speed \displaystyle v(d) = \sqrt{9.8d}, where four times the depth gives twice the speed, along with pendulums and braking distances.
A path through it
Square root graphs first, then context. Cube roots can come at any point, and read most easily after square roots.
Next: restricting a domain to get an inverse, where \displaystyle \sqrt{x} undoes x^2 only on the inputs x \ge 0.