What is a Function in Math Lesson

There are many real-world situations in which it is useful to describe one quantity as it relates to another.

• The amount of a paycheck for an hourly worker depends on the number of hours worked
• The distance traveled by a plane moving at a constant rate of speed depends on the time traveled
• The price paid for fueling a car depends on the number of gallons pumped

Jason works at the local laundromat. He gets paid $19 per hour and gets to choose his weekly schedule. Jason can choose to work any amount, from 0 to 40 hours in 1-hour increments. Additionally, Jason pays no taxes on his income. We could use an equation to model Jason's weekly earnings:
y = 19x, 0 ≤ x ≤ 40, x is an integer
Our equation above shows y as the dependent variable. y is the amount of Jason's weekly paycheck, which depends on x, the number of hours he chooses to work. Since x is chosen by Jason, this is our independent variable. We will also refer to the dependent variable as an "output" and the independent variable as an "input". In other words, we plug in for the input (x) and obtain an output (y). Let's look at this information in a table format:

x » Hours	y » Pay	(x,y)
0	0	(0,0)
1	19	(1,19)
2	38	(2,38)
3	57	(3,57)
39	741	(39,741)
40	760	(40,760)

We often write related information using ordered pairs. We can display a set of ordered pairs which describes Jason's weekly earnings at the laundromat:
{(0,0), (1,19), (2, 38),...,(39,741),(40,760)}
We can refer to this set of ordered pairs or any set of ordered pairs as a "relation". Our relation describes how an input for hours worked relates to an output of pay. When we work with a relation, we usually see:
x is used as the independent variable or input
y is used as the dependent variable or output

Domain and Range

When working with a relation, we will come across the topic of domain and range. The domain of a relation is the set of allowable values that can be plugged in for the independent variable x. The range of a relation is the set of allowable values for the dependent variable y. If we use the above example, we restricted our domain to include only integers from 0 to 40. Therefore, our domain for this relation would be:
domain:{0,1,2,...,38,39,40}
The range would be all of the outputs that are possible. Since we can only plug in certain values for x, our range is also limited.
range:{0,19,38,57,...,741,760}
Let's look at an example.
Example 1: Find the domain and range for the relation.
{(1,3), (7,18), (2, -4), (9,-1)}
Our relation has four ordered pairs. The domain is the set of x-values and the range is the set of y-values.
domain:{1,7,2,9}
range:{3,18,-4,-1}
Example 2: Find the domain and range for the relation.
{(2,-5), (6,-1), (9,3), (7,5)}
Our relation has four ordered pairs. The domain is the set of x-values and the range is the set of y-values.
domain:{2,6,9,7}
range:{-5,-1,3,5}

Function Definition

A function is a special type of relation, where each x-value is associated with only one y-value. When we first encounter functions, we will be given simple examples with a set of a few ordered pairs. Suppose we saw the following relation:
{(3,9), (2,6), (4,3), (7,1)}
Since each x-value is associated with one and only one y-value, this relation is a function.
3 » 9 : an x-value of 3 is associated with a y-value of 9
2 » 6 : an x-value of 2 is associated with a y-value of 6
4 » 3 : an x-value of 4 is associated with a y-value of 3
7 » 1 : an x-value of 7 is associated with a y-value of 1
It may help to view this as an illustration: Now, let's suppose we saw a different relation:
{(9,-2), (4,3), (9,1), (7,-1)}
Since each x-value is not associated with one and only one y-value, this relation is not a function. Notice how the x-value of 9 is associated with two different y-values: -2, and 1. With a function, we want to be able to know with clarity what the value of y is given a specific x-value. If we were to ask, what is the value of y when x is 9? How would you answer? Would you answer -2 or 1? The result is not clear because there is no clear association between an x-value of 9 and a specific y-value. This is why we don't have a function.
9 » -2 and 1 : an x-value of 9 is associated with a y-value of -2 and 1
4 » 3 : an x-value of 4 is associated with a y-value of 3
7 » -1 : an x-value of 7 is associated with a y-value of -1 Let's look at some examples.
Example 3: Determine whether each relation is a function.
{(6,-1), (3,-2), (5, 9), (8,-1)}
Since each x-value is associated with one and only one y-value, this relation is a function. Many students will be confused about this example since there is a duplicate y-value of (-1). This is allowed in the definition of a function. It is clear that an x-value of 6 corresponds to a y-value of -1. It is also clear that an x-value of 8 also corresponds to a y-value of -1. Although it's the same y-value, there is a clear association between each x-value and one y-value.
6 » -1 : an x-value of 6 is associated with a y-value of -1
3 » -2 : an x-value of 3 is associated with a y-value of -2
5 » 9 : an x-value of 5 is associated with a y-value of 9
8 » -1 : an x-value of 8 is associated with a y-value of -1
Example 4: Determine whether each relation is a function.
{(-2,5), (3,-1), (-2,1), (5,8)}
Since each x-value is not associated with one y-value, this relation is not a function. We can see that the x-value of -2 is associated with two different y-values 5 and 1.
-2 » 5 : an x-value of -2 is associated with a y-value of 5 and 1
3 » -1 : an x-value of 3 is associated with a y-value of -1
5 » 8 : an x-value of 5 is associated with a y-value of 8

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