Determine the domain of advanced piecewise functions

The Determine the domain of advanced piecewise functions exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. This exercise practices determining the domain of functions whose expressions are a combination of different types of assignment rules.

Types of Problems[]

There is one type of problem in this exercise:

1. Find the domain of the function: This problem has a function that is written in a rule form. The student is asked to use the rule to find the domain of the function and then make the correct selection from a multiple choice list.

   Find the domain of the function.

Strategies[]

Knowledge of function vocabulary and terminology is encouraged to perform this exercise accurately and efficiently.

1. Students should recall that the domain of a function is the set of all possible inputs, for which the function is defined.
2. For a piecewise function, the possible inputs are all the inputs for which the function has an assignment rule. However, the existence of an assignment rule for an input doesn’t mean the function is defined for that input. This depends on the specific properties of that assignment rule.
3. The domain of a function is the x-values that are allowed to be used. Oftentimes, the domain will be stated as “not” the numbers that are not allowed to be used.
4. Domain questions often come to “not” dividing by zero and “not” taking the even root of negative numbers.

Real-life Applications[]

1. Money as a function of time. One never has more than one amount of money at any time because they can always add everything to give one total amount. By understanding how their money changes over time, they can plan to spend their money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much.
2. Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, and many more. But the important thing is that there is only one temperature output when they measure it in a specific place.
3. Location as a function of time. One can never be in two places at the same time. If they were to plot the graphs of where two people are as a function of time, the place where the lines cross means that the two people meet each other at that time. This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.