Algebra 1 End-of-Course Review

Module Three: Relations and Functions

Venn Diagrams and Sets

In this module, you will learn how Venn Diagrams and Sets are used. Let's review some terms! Select the term below to review its definition.

• Union
  
  In math, the term "union" is everything in two or more sets coming together to make one large set. The term union is represented by a ∪.
• Intersection
  
  In math, the "intersection" of two or more sets represents the items that are the same in both sets. In other words, the intersection of two or more sets is made up of the elements that the sets share. The term intersection is represented using an upside down union symbol which looks like ∩.
• Complement
  
  The complement of set S contains all the elements in the universal set, except those in set S. The complement of set S is written as S′. The little symbol to the right of the S is read "prime," so S′ is read "S prime." Note: this can also be written with the ∼ symbol, and read as "the complement of S."

Now look at a couple of examples to review some of this material.

Relations and Functions

Relations can be represented as a list of ordered pairs, graphed on a coordinate plane, listed in an x-y table of values, or as a mapping.

Using this relation, R = {(2, 3), (1, 2), (4, 2), (2, –1), (–2, 1)}, identify the domain (x-values) and range (y-values).  Recall the following:

• List the numbers separated by commas inside of braces.
• Duplicate values are only listed once.
• List values from least to greatest.

Domain = {-2, 1, 2, 4} Range = {-1, 1, 2, 3}

Is this relation also a function?

Remember that a function is a special relation where each input value (x-value) is used only once. Y-values can be duplicated, but x-values cannot be duplicated. This relation is not a function because the x-value of 2 is repeated. A vertical line test can also be used to determine if a relation is a function. If you can draw a vertical line through the graph of a relation and only pass through one point at a time, then the relation is a function.

To evaluate the function for a particular value, substitute the input value (x-value) and simplify.  The result is the output value (y-value). The input and output values can be written as ordered pairs (input value, output value) or (x, y).

Input value → x Output value → f(x), also known as the y-value

Caution: Sometimes you will be given the output value and be asked to find the input value. Look carefully at the wording.

Here are some more examples that will help you understand domain and range, determining whether a relation is a function, and evaluating functions.