1) Functions map inputs to outputs and are defined by properties like domain, range, graphs, and types including polynomial, rational, radical, and conditional functions. 2) Polynomial functions are defined for all real numbers and have examples, rational functions have restricted domains shown by their graphs, and radical functions' domains depend on whether the radical is even or odd. 3) Functions can have zeroes or roots, and the factor theorem relates the zeros of a function to factors of its polynomial form. Conditional functions output different values depending on the input via piecewise definitions.