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1.1 Characteristics of a Function
Lots of review!
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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.
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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)}
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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.
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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}
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Mapping Diagram The MAPPING DIAGRAM for this relation can be given by:
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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}
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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.
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Example Determine whether each of the following relations is a function. State the domain and range.
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Example Determine whether each of the following relations is a function. State the domain and range. 21 22 45 1 5
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Example Determine whether each of the following relations is a function. State the domain and range. 21 22 45 1 5
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Example Determine whether each of the following relations is a function. State the domain and range. x y 1 5 7 2 3 8
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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.
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Example and Non-Example
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Class/Home work 1.1 Note 1 Worksheet Text questions page 10 #1-6, 7a-c, 9, 11, 12, 15
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