Showing posts with label relation. Show all posts
Showing posts with label relation. Show all posts

Tuesday, July 23, 2013

Relations, Graphs, and Functions

In this section we will study functions in more depth and begin by defining a relation as any set of ordered pairs.
An example of a relation might look like { (2,3), (2,5), (0,1) }. Here we have a relation consisting of three ordered pairs or points on the Cartesian coordinate system. We are familiar with ordered pairs and usually see them denoted as (x, y).  Typically the x-value (the first component) will be the independent variable or input and the y-value (the second component) is the dependent variable or output.
The example { (2,3), (2,5), (0,1) } is NOT a function because the x-value 2 is assigned more than one y-value, namely 3 and 5. For every x-value there can be only one y-value. Next we define the domain as the set of x-values and the range as the set of y-values for which the relation is defined.
Example:
Tip: When looking at a list of ordered pairs, if there are repeating x-values then the relation is not a function.  This usually indicates that there is an input with multiple outputs. (This does not apply to the y-values)

Determine if the relations are functions.  If so state the domain and range.
 
 
Relations can consist of an infinite number of ordered pairs, in which case, making a big list would be impossible. A graph can represent a relation by considering it as a big list of ordered pairs.  Each dot on the graph represents an ordered pair (x, y).
 
Notice that if we can draw a vertical line that intersects the graph twice we will be able to identify one x-value with two corresponding y-values.  Therefore, it can not be a function.
  
Alternatively, if any vertical line crosses the graph only once then it does represent a function.

Use the vertical line test to determine whether or not the graph represents a function.
 
Be prepared to state the domain and range given the graph.  Remember to think of the graph as an infinite set of ordered pairs (x, y).  From the graph, determine the x-values and y-values. First shade in the domain and range then convert to interval notation.
 
It should be clear, at this point, that the circle shown is not a function.  Nonetheless, we still can determine the domain and range of the relation that it represents.


YouTube Video:

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