d. Produce an invertible function from a non-invertible function by restricting the domain.

An invertible function is a function that has an inverse function. Although your students may think that every function has an inverse (after all, we can switch x and y for just about anything and come up with expressions up the wazoo), but switching x's and y's doesn't guarantee a function.

For instance, the function f(x) = x² would have an inverse of , which is not a function. Have the students use the vertical line test if they don't believe you.

For f(x) = x², the domain is all real numbers, but the range is only y ≥ 0. Restricting our domain to x ≥ 0 as well gives us the function defined by the graph below.

By restricting the domain, we've made this non-invertible function an invertible one. Students should be able to tell that the inverse of this function is also a function. Algebraically solving for it or reflecting it across y = x will give us the inverse function, , which passes the vertical line test and is therefore a function.

Students should know not only how to restrict a domain (which is as easy as writing down, "x ≥ 0"), but also which functions will need restricted domains in order to be invertible. Students should also know what the new domain of the function should be.