• Unit 2: Introduction to Functions

    The study of functions is one of the main focuses of this course. In this unit, we will introduce the definition of a function and the special notation used to write equations for functions of two variables. You may find some of the material is familiar if you have already learned about plotting points on the cartesian coordinate plane and writing equations of lines. You will be introduced to a toolkit of functions early in the unit that we will use throughout most of the course, so pay close attention to the characteristics and notation used to define them. You will practice defining important characteristics of functions such as the domain and range. You will practice graphing the many different types of functions that we will focus on throughout the course. Pay close attention to this unit, as it builds the foundation for the course. Enjoy your study of functions!

    Completing this unit should take you approximately 3 hours.

    • Upon successful completion of this unit, you will be able to:

      • summarize the properties of a function, including input and output values, domain, and range using words, function notation, and tables;
      • evaluate a function given an equation, table, or words;
      • identify whether the graph of a relation represents a function;
      • use set builder, inequality, and interval notation to express the domain and range of a function defined by an equation, table, graph, or set;
      • construct the graph of a piecewise-defined function;
      • calculate the average rate of change of a function given a table, graph, or equation; and
      • identify the characteristics of a function given its graph, including behavior over an interval and local and absolute extrema.

    • 2.1: Notation and Basic Functions

      • The Rectangular Coordinate System and Graphs Book

        This section will introduce the cartesian coordinate plane and how to plot points and lines on it. You will also evaluate a two-variable linear equation and learn how a point on the cartesian plane can be a solution to a linear equation in two variables.

      • Defining and Writing Functions Book

        This section introduces the terminology and notation used to define and represent a function using words, function notation, and tables. We will use the concepts and notation introduced in this section throughout the course, so make sure you master them before moving on.

      • Properties of Functions and Basic Function Types Book

        In this section, you will analyze graphs to determine whether they represent a function and be introduced to the graphs of the basic functions. Pay close attention to the basic functions because they will be referred to throughout most of the course.

    • 2.2: Properties of Functions and Describing Function Behavior

      • Finding the Domain of a Function Defined by an Equation Book

        When we purchase something from a retailer, we want to know what kind of payment they will accept. Some online retailers will accept a credit card or Paypal, but not Bitcoin. Similarly, with functions, we cannot always assume that we can evaluate a function using any number as x. For example, what if you are working with the function f(x) = \frac{1}{x} and you try to evaluate the function at x = 0? You cannot divide by zero, so we must let the user know that x cannot be zero, much like how most retailers do not accept Bitcoin. The set of values that can be used for x in a given function is called its domain, and the resulting values that will be output from the function and called the range. In this section, we will find the domain of a function and express it in many ways.
      • Finding Domain and Range from Graphs Book

        In this section, we will find the domain and range of functions given their graphs.

      • Graphing Piecewise-Defined Functions Book

        We can apply the domain and range of functions in piecewise-defined functions. These functions are defined in pieces, and understanding how to define their domains helps us understand their behavior and how to define and describe them.

      • Calculate the Rate of Change of a Function Book

        We will continue exploring functions using equations, tables, words, and graphs by finding average rates of change of functions.

      • Determine Where a Function is Increasing, Decreasing, or Constant Book
        Now that we have more practice graphing and working with equations of functions, we will learn how to describe the behavior of a function over a large interval or by zooming in on a local area where the function's behavior changes.