Week 1 Part 1 - Relationships and Functions

When you have finished this lesson and worked your way through Section 1.1 of the text you should be able to:

• Know the definitions of a relationship and a function.
• Identify a relationship and a function when presented with

  1. Data in a Table
  2. A graph
  3. A formula
  4. A word description

• Have a basic knowledge of functional notation and its meaning
• Know how to use the vertical line test to tell if a graph is a function

Relationships and Functions

We will start with the most general term, a relationship. A relationship exists between two groups or classes of things if: for every value of one quantity there exist one or more values of the other quantity. If you think of this in terms of x and y it means that for every value of x there must be one or more values of y. If this is the true then a relationship exists between x and y.

Next let's look at a more limiting term, a function. You have a function when for every value of x there exists one and only one value of y. For a function to exist several x's may map to the same y. It is not necessary for each x to map to a different y, it is only necessary that every x maps to one value of y.

A relationship and a function can be portrayed in a number of ways. The material below will cover five of the most common possibilities:

• Data shown in figures
• Data shown in tables
• Data portrayed via a graph
• Data displayed using set notation
• An equation that uses functional notation

  The figures shown below illustrate one relation and one function.

  
  

  The same idea can be portrayed with tabular data.

  Shown below are two small tables. In one case (the table on the left) the data represents y² = x (or y equals plus or minus the square root of x). In this case every x maps to two y values so this is a relationship but not a function. The second table on the right represents data from y= x², which is a function. For every x value there is one and only one value of y.

  A Relationship

  X Value	Y Value
  9	-3
  9	+3
  4	+2
  4	-2

  A Function

  X Value	Y Value
  -1	+1
  0	0
  +1	+1

  
  

  Practice Question 1 - Using the data from the table given below is y a function of x?

  x	-3	-2	0	1	2
  y	9	4	0	1	4

  • No, y is not a function of x _____
  • Yes, y is a function of x _____
  • Not enough information to tell if y is a function of x ______

  
  

  The next portrayal of relationships and functions involves displaying the information as a graph.

  To determine if a graph represents a function you can apply the Vertical Line Rule. Basically the rule says that a graph represents a function if every vertical line you could draw will only cut the graph once and only once. If a vertical line will cross the graph more than once then you do not have a function. In the graphs shown below the one on the left represents a relationship while the one on the right represents a function.

  

  Next we will display data using set notation.

  Our data is given as: {(-2, 5), (5, 7), (0, 1), (4, -2)} What we have here are pairs of points (x, y) or (domain value, range value). To determine if the points portray a function you need to look at the x values and decide if each x value maps to one and only one value of y. In the case shown that is true so this is a function.

  If a second set of data is given as: {(9, -5), (9, 5), (2, 4)} you can see that 9 maps to both -5 and +5 so this is not a funciton, it is a relation.
  

  Last we will portray functions and relations using notation of the form y = Q(t) where this is read as y is related t to and the particular relationship is defined by Q(t). This y = Q(t) = 2t +4 would be the case where twice t plus 4 gives us the value of y. Using common notation, if t were equal to 9 this would be written as Q(9) = 2(9) + 4 = 18 + 4 = 22. In other words Q(9) says evaluate the relationship when t = 9. We just don't write it that way we write it as Q(9). Similarly we could write it as Q(t + h) = 2 (a+h) + 4. In this case we have replaced all of the t's on the right side of the equation with their new value which is t+4.

  Practice Question 2 A table of data is shown below: Combining tabular data with the notation discussed above try toanswer the following questions. (answers at the end of this section.)

  x	0	2	4	6	8
  f(x)	-1	3	1	0	1

  Determine the missing values:

  1. f(0) = ? _______
  2. f(x) = 0 _______
  3. f(x) = 1 _______

  Practice question 3 - Match the following statements to the figures shown below:

  1. We purchased the machine for $10,000. Two years later it was worth $5,000.
  2. Starting from nothing ($0) our sales went to $10,000 per month is one year.
  3. We bought the stock for $5,000. One year later it was worth $10,000.
  4. We purchased the lot for $10,000 thinking the area would grow. However, problems caused the property value to fall to 1/2 its original price only two years after our purchase

  .

  Practice question 4 -. Determine which variable to put on the x axis and which to put on the y axis for the following situations:

  1. Costs (C) were a function of the quantity (Q) of the product that we produced. C goes on the _____axis. Q goes on the _____axis.
  2. Our bank balance (B) was a function of the day of the month (T). B goes on the ____axis. T goes on the ____axis.
  3. The area (A) of a square is a function of the length of its sides (S). A goes on the ____axis. S goes on the ____axis.

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