Relations & Functions A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )

Relations & Functions A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height ) This set { 15 , 65 , 21 , 70 , 18 , 73 , 16 , 62 } is a relation.

Relations & Functions A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height ) This set { 15 , 65 , 21 , 70 , 18 , 73 , 16 , 62 } is a relation. The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 }

Relations & Functions A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height ) This set { 15 , 65 , 21 , 70 , 18 , 73 , 16 , 62 } is a relation. The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 } The range of a relation are the output or y – values { 65 , 70 , 73 , 62 }

Relations & Functions ** ALL SETS OF ORDERED PAIRS ARE RELATIONS A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height ) This set { 15 , 65 , 21 , 70 , 18 , 73 , 16 , 62 } is a relation. The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 } The range of a relation are the output or y – values { 65 , 70 , 73 , 62 } ** ALL SETS OF ORDERED PAIRS ARE RELATIONS

Relations & Functions A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once.

Relations & Functions A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once. This set 0 , 2 , 1 , 3 , 2 , 4 , 3 , 5 is a function. No x – values repeat.

Relations & Functions A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once. This set 0 , 2 , 1 , 3 , 2 , 4 , 3 , 5 is a function. No x – values repeat. This set (1, 5 , −1 , 1 , 0 , 0 , −1 , 7 , 2 , 4 } IS NOT a function because x – value of – 1 repeats or “shows up” more than once.

Relations & Functions Example : Does the table below represent a function ? Laps ( x ) 3 8 5 6 4 Minutes ( y ) 9 14 15

Relations & Functions Example : Does the table below represent a function ? Laps ( x ) 3 8 5 6 4 Minutes ( y ) 9 14 15

Relations & Functions Example : Does the table below represent a function ? A quick scan of the x – values show none of them repeating so YES, this table is a function. Laps ( x ) 3 8 5 6 4 Minutes ( y ) 9 14 15

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. Here is a linear function : Test if it is a function with the vertical line test…

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. Here is a linear function : Test if it is a function with the vertical line test… No vertical line we draw crosses the graph more than once so this graph represents a function.

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. Does this graph represent a function?

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. Does this graph represent a function? No, we only need to find one line that intersects the graph more than once.

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. This function has some opened and closed circles. The point ( 5 , 4 ) has an open circle which is not part of the graph. The points ( - 8 , 2 ) and ( 5 , - 2 ) are closed circles and considered part of the graph.

Relations & Functions A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function. Even though the vertical line appears to cross more than once, one of the points is open and not part of the graph. So this graph is a function.

Relations & Functions Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets.

Relations & Functions Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets. EXAMPLE : Map the given function −2 , 2 , −1 , 5 , 0 , 8 , ( 1 , 11 )

Relations & Functions Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets. EXAMPLE : Map the given function −2 , 2 , −1 , 5 , 0 , 8 , ( 1 , 11 ) DOMAIN RANGE −2 −1 1 2 5 8 11 This mapping shows a function as all domain values are paired with only one range value…

Relations & Functions EXAMPLE : Does this mapping show a function ? DOMAIN RANGE 10 −5 3 6 −2 5

Relations & Functions EXAMPLE : Does this mapping show a function ? DOMAIN RANGE 10 −5 3 6 −2 5 No, one domain value maps into more than one range value… - 2 shares – 5 and 0

Relations & Functions EXAMPLE # 2 : Does this mapping show a function ? DOMAIN RANGE −2 2 4

Relations & Functions EXAMPLE # 2 : Does this mapping show a function ? DOMAIN RANGE Yes, even though -2 and + are pointing to the same range value, each domain value is still paired only one time −2 2 4