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Relations & Functions—NOTES Objectives: I can identify a function from a written description, table, graph, rule, set of ordered pairs, and/or mapping. Vocabulary Relation: A set of ____________________________. Function: A relation in which each member of the _____________ is paired with exactly one member of the _________________. Domain: The set of ______________ values. Range: The set of __________________ values. Independent Variable: These values are chosen and do not depend on the other variable. In a set of ordered pairs, the ____________________ is the independent variable. Dependent Variable: This value depends on the input value/independent variable because it changes when the input value changes. In a set of ordered pairs, the ___________________ is the dependent variable. Key Concepts Determining if a Relation is a Function A relation is a function if each ____________________ is matched up with ONLY ONE _____________________. To determine from a list or table. Does a number in the domain match up with two different numbers in the range? No—Then the set of ordered pairs is a function. Yes—Then the set of ordered pairs is not a function because one x-value has two different y-values. Examples: {(2,1), (4,3), (5,4), (9,7)} x 2 5 9 2 y 1 4 7 3 Determining if a Relation is a Function from a Graph To determine if a relation is a function when the ordered pairs have been graphed, you can apply the _________________________________________ to the graph of the relation. Place a pencil at the left of the graph along the _______________________. If, for each value of x in the domain, the pencil passes through only one point of the graph, then the graph represents a function. Example: Function Notation A function that is written as an equation can also be written in a form called _______________________________________. Equation Function Notation y=4x f(x) = 4x The number in the ( ) will tell you what to sub for x in the problem. Examples: Find f(3) if f(x) = 5x. Find f(4) if f(x) = 8x Relations & Functions—NOTES Objectives: I can identify a function from a written description, table, graph, rule, set of ordered pairs, and/or mapping. Vocabulary Relation: A set of ____ordered pairs________________________. Function: A relation in which each member of the __x-coordinate______ is paired with exactly one member of the _y-coordinate__________. Domain: The set of ____x___ values. Range: The set of _________y_________ values. Independent Variable: These values are chosen and do not depend on the other variable. In a set of ordered pairs, the ____x-coordinate_______ is the independent variable. Dependent Variable: This value depends on the input value/independent variable because it changes when the input value changes. In a set of ordered pairs, the _____ y-coordinate _________ is the dependent variable. Key Concepts Determining if a Relation is a Function A relation is a function if each ___ x-coordinate _________ is matched up with ONLY ONE ____________y-coordinate _______________. To determine from a list or table. Does a number in the domain match up with two different numbers in the range? No—Then the set of ordered pairs is a function. Yes—Then the set of ordered pairs is not a function because one x-value has two different y-values. Examples: {(2,1), (4,3), (5,4), (9,7)} Because there is only one yvalue for every x-value, this IS a function! x y x 2 5 9 2 x y 2 1 y 1 4 7 3 2 1 4 3 5 4 5 4 9 9 7 7 3 Because the xvalue of 2 has yvalues of both 1 and 3, this IS NOT a function! Determining if a Relation is a Function from a Graph To determine if a relation is a function when the ordered pairs have been graphed, you can apply the _________pencil line test____________ to the graph of the relation. Place a pencil at the left of the graph along the __x-axis___________. If, for each value of x in the domain, the pencil passes through only one point of the graph, then the graph represents a function. Example: This graph does represent a function because the pencil does not touch two points at the same time as you roll it along the x-axis. Function Notation A function that is written as an equation can also be written in a form called __function notation______. Equation Function Notation y=4x f(x) = 4x read as F of X The number in the ( ) will tell you what to sub for x in the problem. Examples: Find f(3) if f(x) = 5x. Find f(4) if f(x) = 8x f(x) = 5x f(3) = 5(3) or 5 x 3 f(3) = 15 f(x) = 8x f(4) = 8(4) or 8 x 4 f(4) = 32