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Ohio Open Ed Collaborative
Mathematics, Tmm0022
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Introduction - Cartesian Plane, Relations and Functions, Function Notation

Relations and Functions

Many things in the world around us can be described in terms of relationships and functions that operate on inputs to give meaningful outputs. The set of inputs is called the domain and the set of ouputs is called the range of the underlying relationship/function. Many complex objects are the "sum" of individual basic components. It is possible to understand how such complex objects work by understanding how the individual components work. 

In this module we will learn about mathematical functions that form the foundation of calculus. The discussion centers around what a function is made of, how it operates, and its applications. By the end of this module, you should be able to

  • Identify domain and range of a function
  • Describe and notate a function using letters
  • Evaluate a function
  • Find the average rate of change in the value of a function with respect to changes in input values
  • Find the difference quotient
  • Graph a function  or a relationship
  • Perform Vertical Line Test to tell if a graph represents a relationship or a function.
  • Study transformations of functions


Sets of Real Numbers and the Cartesian Coordinate Plane

A quick review of set notions pertaining to intervals, their unions and intersections, and the Cartesian plane will be helpful to students. Also, notions of symmetry of functions and relationships can be explored here.

This is a review of some basic notions of sets and the Cartesian plane, mid-point and distance formulas. You will use set notation to describe domains of functions and relationships in terms of intervals.


A relation is a set of ordered pairs (x, y). Such a set can be described verbally or graphed in the plane. It is much easier to describe a relation in terms of an equation between the variables when they take on numbers. For example the statement

Picture A is the graph of a circle with center (4, 5) and radius 2.

can be described by the equation

\((x - 4)^2 + (y - 5)^2 = 4\)

where x and y are real numbers. It is easy to show that x varies in the interval [2, 6] and y varies in the interval [3, 7].

We can use an equation describing a relation to test for symmetry and for finding the intercepts, which are coordinate points of intersection of the graph with the x and y axes.

Learning Objectives:

  • Graph a relation
  • Describe a relation
  • Graph an equation
  • Find intercepts of a relation
  • Perform test for symmetry

Functions - Introduction and Notation

A function is a combination of one or more parts/operations that act on an input in  a praticular way and give exactly one output. A function is a relation but a relation need not be a function. A function may return the same output value for different input values. On the other hand a relation may return different outputs for one input value. In this section we will learn about common features of functions including domain and range, notation, and evaluation.

Learning Objectives:

  • Describe a function
  • Determine the domain and range of a function
  • Test if a a graph represents a function - Vertical Line Test
  • Use notation to represent a function
  • Evaluate a function
  • Modeling with a function