Domain, range, piecewise-defined functions
• Use the notions of domain and range (F-IF.A.1, F-IF.B.5$^\star$).
• Interpret a graph of a piecewise-defined function (F-IF.C.7b$^\star$).
• Graph step functions (F-IF.C.7b$^\star$).
In this section, students are introduced to the notions of domain and range. They work with graphs of two types of functions that may not be familiar: continuous piecewise-defined functions and step functions. Because the ranges of step functions are discrete, students may find the notion of range especially salient in the case of step functions.
WHAT: A function that represents a context is described, and students are asked to state a reasonable domain and range for the context.
WHY: The purpose of this task is to get students thinking about the domain and range of a function representing a particular context. Often when a function is being used to model a context, the expression for the function has a larger domain and range than is reasonable for the context. Asking students to focus on a function for which there is no formula focuses attention on the context itself. In order to complete this task, students will need to reason abstractly about the quantities in the context (MP2).
WHAT: The function $f(x) = 2/(x – 3)$ is defined. Students evaluate $f(11)$ and attempt to evaluate $f(3)$. Then students are asked to list the algebraic operations needed to evaluate an input and describe restrictions that each operation places on the domain of the function.
WHY: The purpose of this task is to introduce the idea of the domain of a function by linking it to the evaluation of an expression defining the function. By thinking through the evaluation step by step, students isolate the exact point where a given input results in an undefined output. In part (d), any domain that excludes $x=3$ is possible. It is conventional when given a function defined by an expression to take the domain to be the largest possible, but it is worth pointing that this is a convention, not a mathematical fact. As students gain a mature understanding of functions they learn that the domain is something that is specified when you define the function, it does not come already attached. The function in this task can be broken down into two simple operations. More complicated variations are possible. Students must make use of structure to evaluate operations in the correct order. Also, stepping through the operations like this lays the groundwork for students to be able to make use of structure in future work (MP7).
WHAT: Students are asked to graph the cost of parking in a garage where the cost increases each half-hour or fraction thereof.
WHY: This task provides an example of a step function arising from a real-world context. The idea that the price “jumps” \$0.50 each half-hour or fraction thereof gives rise to why a single linear function is not an appropriate model for this problem (MP4). This task lends itself to a discussion about domain and range and why the graph requires “hollow dots” and “solid dots” to indicate jumps.
WHAT: Students are presented with a piecewise-defined graph that shows the balance in a bank account over a seven-day period. The graph is shown as a continuous function. Because the bank account balance changes discontinuously, the graph of a step function is a more accurate representation.
WHY: This task provides students with an opportunity to explain why the function shown in the graph is not the most appropriate model for the given context (MP2). The second part of this task is open-ended and allows for various models (MP4) (a step function or bar graph are two possibilities). This could lend itself to a class discussion where students explain why their model best represents the given situation (MP3).