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2.1 Functions and their Graphs Function Identification Essential Questions How can functions be identified? How do functions differ from relations? How are linear functions distinguished from nonlinear? Relations and Functions A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. The set of output values is called the range. A relation as a function provided there is exactly one output for each input. A vertical line test can be used to determine if a graph is a function. It is NOT a function if at least one input has more than one output. A linear function is one that forms a line when it is graphed (remember: linear = line). A nonlinear function does not form a line. Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 Notice the set notation!!! Domain = {-3, 1,4} Range = {3,-2,1,4} 4 Function? No: input 1 is mapped onto Both -2 & 1 Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output FUNCTIONS Many to One Relationship One to One Relationship X 1 2 3 Y 0 0 0 X 1 2 3 Y 5 7 9 Vertical Line Test You can use the vertical line test to visually determine if a relation is a function. Slide any vertical line (pencil) across the graph to see if any two points lie on the same vertical line. If there are not two points on the same vertical line then the relation is a function. If there are two points on the same vertical line then the relation is NOT a function (b) (a) (c) (d) Graphing and Evaluating Functions Many functions can be represented by an equation in 2 variables: y=2x-7 An ordered pair is a solution if the equation is true when the values of x & y are substituted into the equation. Ex: (2,-3) is a solution of y=2x-7 because: -3 = 2(2) – 7 -3 = 4 – 7 -3 = -3 In an equation, the input variable is called the independent variable. The output variable is called the dependent variable and depends on the value of the input variable. In y=2x-7 ….. X is the independent variable. Y is the dependant variable. The graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation. Graphing an equation in 2 variables 1. Construct a table of values 2. Graph enough solutions to recognize a pattern 3. Connect the points with a line or curve Graph: y = x + 1 Step2: Step 1 Table of values Function Notation By naming the function ‘f’ you can write the function notation: f(x) = mx + b “the value of f at x” “f of x” f(x) is another name for y (grown up name) You can use other letters for f, like g or h Function Function Notation: f(n) = n + 5 Output Input Function Function Notation: f(n) = n + 5 Rule for Function