To define a function we need a set of inputs (called the domain) and some sort of rule for turning inputs into outputs. The set of outputs is called the range of the function.

Here are some rules which tell us what will happen to a given input value, \(x\).

rule A x maps to x squared minus 2, rule B x maps to x minus x cubed, rule C x maps to 4 over x squared minus 4, rule D x maps to the square root of x plus 2

For each rule,

  • Define a function by choosing a domain. Now define another function by choosing a different domain.

  • For each function you have defined, state the corresponding range of the function.

  • Sketch graphs of the functions you have defined.

For each rule,

  • What is the largest possible domain you can define within the real numbers?

  • What is the smallest possible domain that will define a function with the largest possible range?