RELATIONS AND FUNCTIONS: DEFINITIONS

1. the domain set of C = {( 2, 5), (2, 6), (2, 7)} 2. the range set of E = {(3, 3), (4, 4), (5, 5), (6, 6)} 3. the range and domain of F = {(x, y ) | x + y =10} 4. the range and domain of P = {(x, y) | y = 3}

1. {2} 2. {3, 4, 5, 6} 3. domain = range = {all real numbers} 4. domain = {all real numbers}: range = {y: y = 3}

1. domain 2. function 3. input 4. output 5. range 6. relation

4. the y-value of a function. 6. any set of ordered pairs (x, y) that are able to be graphed on a coordinate plane. 2. a relation in which every input value has exactly one output value. 3. the x-value of a function. 5. the second element of a relation or function; also known as the output value. 1. the first element of a relation or function; also known as the input value.

Example 1:

A = {(6, 2), (2, 6), (4, 1), (4, 0)} Set A is a relation because it is a set of ordered-pair numbers. Set A is finite with four members. Although the set of ordered pairs is considered to be a relation, it is not considered a function since it has x-values that are repetitive. Input: (6, 2, 4) Output: (2, 6, 1, 0)

Which of the following statements best represents the relationship between a relation and a function.

A function is always a relation but a relation is not always a function.

Relations and Functions: Definitions

A person can only be in one place at one time. At any one moment, a person only measures one thing, they can't be both one hundred fifty pounds and 90 pounds. That isn't possible. A student's overall class grade can be passing or failing. It can't be both. The concept of function is one that is key to mathematics and is key in our world. The idea of one input having only one output is really commonplace. [Function: for every input there is only one output. Example. Position 1 Weight = 110. Position 2. Weight = 115. Semester. Fall or Fail. Fall: pass Spring: pass] The goal of this section is to give you some more exposure to identifying if a set of data could be described as a function or more generally as a relation. A relation can be any relationship between different quantities. Inside this collection of relations, there are some special relations called functions. [Large circle labeled A relationship. The circle also contains a smaller circle with a sentence: For every input there is only one output.] Examples of functions are things like those described before where at one moment there is only one possible output. Mathematically, a function is defined as a relation where for each input, x value, there is only one output, y value, related to it. Here you are given elements of a relationship described by the ordered pairs zero two, four nine, six 12 and seven two. Notice that for each x value, the first number in the ordered pair, there is only one y value that relates to it. This is a function. [Examples of function: {(0, 2), (4, 9), (6, 12), (7, 2)}] But be careful here. There are two different inputs that both relate to the output of 2. This is not a problem. It is still functional. If you consider a scenario it would be like a person going on a round trip in a car. If they started at time zero in position two and drove for seven units of time, going through other positions but ending back at position two, where they started. This is very possible and thus functional. Something that is not functional would be like the set of ordered pairs zero two, four nine, five six, five eight and seven two. From the sets we can see that when the input is 5 the output is both six and eight. Recall the example of a person weighing 150 pounds and 90 pounds at the same time. [How could this happen? Not a chance, so it's not a function] If we were to try and make a model of this with a car like before we would run into a problem. It wouldn't be functional to say that a car could be in two places at once, so this set would fall into the larger group of relations rather than the smaller circle of functions.

Example 2:

B = {(x, y) | x + y = 5} Set B is a relation; it is also an infinite set because it does not contain a definite set of x and y values. From the infinite set, we conclude that the domain and range of the linear function is all real numbers. Remember: Set A is designated by the list method and Set B by the rule method. A B = {(4, 1)}; A B is an infinite set.

Example 3:

C = {(2, 5), (6, 1), (8, 2), (9, 5), (4, 10)} Set C is a function because each first element is listed only once. Domain: (2, 6, 8, 9, 4) Range: (5, 1, 2, 10)

Example 4:

D = {(2, 6), (5, 1), (2, 7), (8, 3)} Set D is not a function because first element 2 has two different values for second elements, 6 and 7. However, set D is a relation because any set of ordered pairs is a relation.

Example 6:

Determine if the relation is a function. If the relation is a function, name the domain and range. {(-1, 7) (3, 0) (1, 4) (-5, 9)} This relation is a function because as you can see, the x-values are not repetitive. In case you are wondering why the x-values -1 and 1 are not considered repeating—they are the same digit 1, however, because one of the digits has a negative sign, they have different numerical values. Domain: {-1, 3, 1, -5} Range: {7, 0, 4, 9}

Select the domain and range of F. F = {(x, y ) | x + y = 10}.

Domain: {10} Range: {10}

If the relation is a function, list the domain and range. If the relation is not a function, choose "not a function". C = {(9, 1) (8, -3) (7, 5) (-5, 3)}

Domain: {9, 8, 7, -5} Range: {1, -3, 5, 3}

Example 5:

E = {(x, y) | x - y = 8} Set E is a function because for any value chosen for x, you will get a single value for y. Notice that Set E uses the rule method for designating the set. This linear function has a domain and range of all real numbers since it is considered to be an infinite set.

*

The domain and range can be listed in the order the numbers are given or can be rearranged numerically, lowest number to highest number. It is not necessary to list repeated numbers.

DEFINITIONS

You may recall from your study of two-variable equations that the graphs of those equations are in a two-dimensional plane. Each point on the graph has an ordered pair that describes its location. Each ordered pair "solves" the equation. You can think of all these ordered pairs as a set of ordered pairs called a relation. In the case of an equation, the set would have an infinite number of ordered pairs. This would make it impossible to use the list method for sets. To describe the relation set, the best way is to use the rule method and the equation itself as the rule. For finite sets of ordered pairs, an equation is hard to find; however the list method works nicely!

function

a function is a relation such that for each first element (x-value, input) there exists one and only one (unique) second element. Another way to say this is that none of the ordered pairs have a repetitive x-value. That is, every first element (x-value, input) is used only once. Although every function is considered a relation, not all relations are considered to be a function

relation

a relation is any set of numbers that are able to be graphed on a coordinate (x, y) plane. A relation can be a function, but is not always a function

Select either relation (if the set is a relation but not a function), function (if the set is both a relation and a function), or neither (if the set is not a relation). A = {(1, 2) (2, 2) (3, 2) (4, 2)}

function

Select either relation (if the set is a relation but not a function), function (if the set is both a relation and a function), or neither (if the set is not a relation). F = {(x, y ) | x + y = 10}

relation

Select either relation (if the set is a relation but not a function), function (if the set is both a relation and a function), or neither (if the set is not a relation). input -1 0 -1 -8 9 output 0 1 2 4 8

relation

domain

the first element (x-value) of a relation or function; also known as the input

input

the input is the x-value of a relation or function

output

the output is the y-value of the relation or function

range

the second element (y-value) of a relation or function; also known as the output.