TOPIC 3: Functions A. Functions vs. Relations

Do you remember this? ab Find the rule The rule is: 3 a = b The question could also look like this: Find the rule for the following ordered pairs: (2,6); (5,15); (10,30); (100,300) The rule is: 3x = y3x = y

3 x = y Notice that we call the first number of each pair x, the INDEPENDENT VARIABLE and the second number of each pair y, the DEPENDENT VARIABLE. ( x, (2, (5, (10, (100, y ) 6) 15) 30) 300) You will remember that we call them ordered pairs and we can also use them to graph points in the coordinate system (Cartesian plane) DEPENDENT VARIABLE INDEPENDENT VARIABLE

For example we could graph the first two ordered pairs: (2,6) (5,15) (2,6) and (5,15)

Any set of ordered pairs is called a relation Age (years) Height (centimeters) Example: The table presents the ages and heights of members of a large family. We could represent this information as a set of ordered pairs: {( 1, 58); (7, 106); (13,157); (15,165); (38, 165); (43, 176 )}

{(1, 58),(7, 106), (13,157), (15,165), (38, 165), (43, 176)} When we are using sets, the correct notation is... To write the elements of the set between brackets and to separate the elements by using commas { } ,,,,, CORRECT NOTATION

{(1, 58),(7, 106), (13,157), (15,165), (38, 165), (43, 176)} This type of information is representing a RELATION between two sets of data, namely the age and the height Note that in (7, 106) ≠ (106, 7), CLEARLY the first ordered pair is the one that gives the CORRECT RELATION between the age and the height – (7, 106)

a 7 year old is not the same as an 106 year old with a height of 106 cm with a height of 7 cm! 106 cm 7 cm (7, 106) ≠ (106, 7)

{(1, 58),(7, 106), (13,157), (15,165), (38, 165), (43, 176)} The set of all the first elements of the ordered pairs (the ages) is called the The set of all the second elements of the ordered pairs (the heights) is called the DOMAIN RANGE {1, 7, 13, 15, 38, 43} SO THE DOMAIN IS : AND THE RANGE IS : {58, 106, 157, 165, 176}

{(4, -2), (4, 2), (9, -3), (9, 3}3} EXAMPLES: State the domain and the range of each relation 1. {(-3, 0), (4, -2), (2, -6)} DOMAIN: RANGE: {-3, 4, 2} {0, -2, -6} 2. {(4, -2), (4, 2), (9, -3), (9, 3} DOMAIN: RANGE: {4, 9} { -2, 2, -3, 3} When elements are repeated in a set we only write then once! ONCE! REPEATED CORRECT NOTATION

Now the question is, when are relations functions? Domain Range with exactly one element in the range A Function is a relation in which each element of the domain is paired exactly one each

Range A Function is a relation in which each element of the domain is paired with exactly one element in the range exactly one each Key Words here are: and Domain

Range exactly one each EXAMPLES: So in this example since there is an element of the domain that is NOT paired with an element of the range then this is NOT a function, only a relation Is this relation a function? NO! YES!

DomainRange exactly one each EXAMPLES: How about this relation, is it also a function? This relation is not a function! YES! NO!

Domain Range exactly one each EXAMPLES: Is this relation a function? YES! YES! This relation is a function!

Height Range Age Domain Remember that the ages are the domain and the heights are the range Let’s go back to the example with the ages and heights of a family. {( 1, 58); (7, 106); (13,157); (15,165); (38, 165); (43, 176 )} These were the ordered pairs depicting the relation

Heights Range Age Domain eachexactly one DomainRange So in this case, the relation is a function YES! YES!

Range Domain each exactly one Domain Range The relation is NOT a function YES! NO! Is this relation a function? EXAMPLE: {(4, -2), (4, 2), (9, -3), (9, 3)}

157 By now we have figured out that there is another way of defining A Function is a set of ordered pairs in which no two pairs have the same first element. This definition is very useful when we want to use a graphical approach. In other words, all the ordered pairs have different x-coordinates.

then the line is called x = 3 Remember... Vertical lines are called x = a, where a is any real number. For example if a vertical line goes through the number 3 on the x-axis X = 3

{( 1, 58); (7, 106); (13,157); (15,165); (38, 165); (43, 176 )} is a set of ordered pairs in which no two pairs have the same first element. The Function or shown and said in a different way... There is no vertical line that can be graphed that will pass through more that one point on the graph. One more time let’s go back to the example with the ages and heights of a family. X=1 X=7 X=13 X=15 X=38 X=43

Is this relation a function? EXAMPLE: {(2, -3), (3, 5), (4, -5), (4, 9), (5, 6), (5, -8)} X = 4

The relation is NOT a function X = 4 Since there is a vertical line, x = 4 that is passing through more that one point on the graph This is called the VERTICAL LINE TEST

So far we know FUNCTION A set of ordered pairs in which no two pairs have the same first element is a Any set of ordered pairs is called a The set of all the first elements of the ordered pairs is called the The set of all the second elements of the ordered pairs is called the DOMAIN RANGE RELATION A relation in which each element of the domain is paired with exactly one element in the range is a

3A functions-relations Presumed knowledge: concept of relation mappings Topic 3: Functions and Equations concept of a function domain, range 3A