Functions

Given two sets it is possible to take elements of one set and assign elements of the second set to them. For example Given the two sets A={ The students in our class } and B={A,B,C,D,F} I can assign to each student a letter based on their final grade. Such a correspondence between two sets is called a Relation. We could also do the same with sets of numbers. Consider the set A={1, 2, 4, -1, 5, 0} and B={5, 7, 11, 1, 3, 13}, we can assign to each element of the first set, an element of the second set.

In this case the rule for our relation is y = 2x + 3 (each element of the first set is first doubled and then increased by 3). We can also show this relation by using coordinate notation {(1,5) , (2,7) , (4,11) , (-1,1) , (5,13) , (0,3)}. We could also plot these coordinate and represent the relation as a graph. Graphs, sets and formulas are a few of the ways we can represent a relation. If a relation satisfies certain conditions we say that the relation is a function.

A function is a relation that assigns to each element of the first set (the domain) one and only one element of the second set (the range).

The relation y = 2x + 3 defined above is a function. Since each element of the domain (set A) is assigned a unique element in the range (set B).
Now consider the sets A={ 1, 4, -1, 5, 0} and B={5, 7, 11, 3, 13} and the following relation:

The above relation is not a function because it assigns two different values of the range to the the domain element "1" .

Example 1) Is the given relation a function? If so find the domain and the range.

The first relation is a function because each element of the domain {3,-1,5,7} is assigned a unique element of the range {4,5,7}. Notice that both 5 and 6 are assigned the same element "7". In a function this is not a problem, we only have to make sure that each domain element of the first set does not have two different range elements assigned to it. The second relation is not a function because 3 has two different elements (4 and 5) assigned to it.
Going back to the example of students and grades: I can assign two different students the same grade, but can not assign two different grades to the same student.

Example 2) Which of the following represents a function?

The equation y = 3x² - 5 does represent a function. Choose any value for x, plug it in, and you get out one y-value. Let x = 5, then y = 3(5)² - 5 = 3(25) - 5 = 75 - 5 = 70.
In contrast the equation y² = x, does not represent a function. Any value of x, except 0, gives us two values of y. For example, let x = 9, then y could be either 3 or -3. One element of the domain, 9, is assigned to 2 elements of the range, 3 and -3, so this equation does NOT represent a function.

Now that we know what a function is we need to learn about

Function Notation

Practice problems

Consider the following equations, which represent functions?

1. y = 3x + 7

2. | y | = 3x

3. 5x² - 2y = 10

4. y² - x = 5

Answers

Solutions

1. y = 3x + 7 Function

(each value of x gives one value of y)

2. | y | = 3x Not a Function

(let x = 1, then we have |y| = 3, and when we solve for y we get y = 3 and y = -3. Thus one element of the domain, 1, gives two elements of the range, 3 and -3, and we do not have a function.)

3. 5x² - 2y = 10 Function

(each value of x gives one value of y)

4. y² - x = 5 Not a Function

(let x = 4, then we have y² - 4= 5, or y² = 9 and when we solve for y we get y = 3 and y = -3. Thus one element of the domain, 4, gives two elements of the range, 3 and -3, and we do not have a function.)

Back to the top