1.1 Characteristics of a Function Lots of review!

Graphing Basics The height of a tree is related to the diameter of its trunk. The table shows the height and the diameter of six maple trees. Graph the following data. Note: Label the x and y axis with variables and units and include a proper title.

Definitions A RELATION is: a rule that associates each element “x” in a set “A” with one or more element(s) “y” in a set “B”. The result is a set of ordered pairs (x, y). In the example above, set A is the diameter of the tree trunk while set B is the height of the tree. The set of ordered pairs for this relation is: {(112, 27), (120, 28), (122, 29), (122, 30), (132, 31), (140, 33)}

Variables The INDEPENDENT VARIABLE is: Diameter of the tree measured in cm. This is also called the manipulated variable. The DEPENDENT VARIABLE is: Height of the tree measured in m. This is also called the responding variable. It is clear that height depends on diameter, not vice versa.

Domain and Range The DOMAIN of a relation is the set of 1st coordinates in the ordered pairs (set A) The RANGE of a relation is the set of 2nd coordinates in the ordered pairs (set B) The domain of the relation above is: D = {112, 120, 122, 132, 140} The range of the relation above is: R = {27, 28, 29, 30, 31, 33}

Mapping Diagram The MAPPING DIAGRAM for this relation can be given by:

Example Months: Write a set of ordered pairs in the form (l, d), where l is the number of letters in the name of the month and d is the number of days in the month in a leap year. {(7, 31), (8, 29), (5, 31), (5, 30), (3, 31), (4, 30), (4, 31), (6, 31), (9, 30), (7, 31), (8, 30), (8, 31)} Write the domain and range for this relation. D = {3, 4, 5, 6, 7, 8, 9) R = {29, 30, 31}

Functions A FUNCTION is a special type of relation in which there is only one value of the dependent variable for each value of the independent variable. In other words, for every x-value, there is only one y- value. Simply put, a function is a relation whereby the x-value DOES NOT REPEAT ITSELF. Each value of the domain has ONE value of the range.

Example Determine whether each of the following relations is a function. State the domain and range.

Example Determine whether each of the following relations is a function. State the domain and range. 21 22 45 1 5

Example Determine whether each of the following relations is a function. State the domain and range. 21 22 45 1 5

Example Determine whether each of the following relations is a function. State the domain and range. x y 1 5 7 2 3 8

Vertical Line Test (VLT) To determine whether a graph represents a function, use the VERTICAL LINE TEST (VLT). VERTICAL LINE TEST – If no two points on the graph of a relation lie on the same vertical line (along a ruler for example), the relation represented by the graph is a function. Use this explanations for given graphs! If data or an equation are given, sketch it and use VLT OR use the one output for every input definition instead.

Example and Non-Example

Class/Home work 1.1 Note 1 Worksheet Text questions page 10 #1-6, 7a-c, 9, 11, 12, 15