Functions

Objectives Define a relation Define a function Define the domain and range of a function Evaluate a function Graph a function

Set of Ordered Pairs The table shows the number of medals won by United States athletes during five Winter Olympics. The following is a set of ordered pairs: {(x, y) | x = year, y = number of medals} {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)}

Domain and Range of a Relation Relation: set of ordered pairs {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)} Domain: set of all first components of a relation {1992, 1994, 1998, 2002, 2006, 2010} Range: set of all second components of a relation {11, 13, 34, 25, 37}

Example Given the relation {(–2, –5), (4, 7), (8, 9)} Find its domain {–2, 4, 8} Find its range {–5, 7, 9}

Function Function A function is a set of ordered pairs (a relation) in which to each first component, there corresponds exactly one second component. Is this a function? {(1, 2), (2, 4), (3, 6), (4, 8), (, 10)} Yes {(2, 4), (3, 9), (4, 16), (5, 25), (6, 36)} {(36, 6), (25, 5), (25, -5), (16, 4), (9, 3)} No {(1, 6), (2, 5), (3, -5), (4, 4), (4, 3)}

Y as Function ox X Given the equation: y = 3x – 4 Since the equation describes a set of ordered pairs, it describes a relation. Since, for each value of the domain, there is only one value of the range, the equation also describes a function.

Y as Function of X y is a Function of x Any equation in x and y where each value of x determines exactly one value of y is called a function. In this case, we say that y is a function of x. y = 3x – 4 Variable y is a function of x. Variable x is the independent variable. Variable y is the dependent variable.

Function Comments Function is always a relation, but relation is not always a function. E.g., {(2, 1), (2, 3)} is a relation, but not a function Another view of a Function x (independent y (dependent variable) variable) 2 -10 -1 -7 0 -4 1 -1 2 2 y = 3x - 4 (function)

Example Determine whether the equations define y to be a function of x. y = x2 x = y2

Solution y = x2 The table shows that for each value of x, there is exactly one value of y. So, the relation is a function.

Solution x = y2 We construct a table of ordered pairs for the equation x = y2. Because y is squared, it will be more convenient to substitute values for y and compute the corresponding values for x. The table shows that for each value of x, there are more than one value of y. Thus, the equation does not describe a function.

Function Notation Write y = 2x – 3 as f(x) = 2x – 3. Read this as: “f of x = 2x – 3.” The notation y = f(x), read “y equals f of x,” means that the variable y depends on the value of x.

Examples Let f (x) = 2x – 3. Find the following f (3) f (–1) f (0) the value of x for which f (x) = 5. f(x) = 2x - 3 5 = 2x – 3 8 = 2x x = 4

Graph of a Function Given: f (x) = | x | Graph the function Determine the domain and range

Domain and Range Domain—set of all real numbers D = {x | x is a real number} Range—set of all positive numbers and 0 R = {y | y ≥ 0}

Another View of Function x y 0 1 4 9 -2 -1 0 1 2 3 Relation x y -2 -1 0 1 2 3 0 1 4 9 Function

Determine whether a graph represents a function A vertical line test can be used to determine whether the graph of an equation represents a function. If any vertical line intersects a graph more than once, the graph cannot represent a function, because to one number x, there would correspond more than one value of y. The graph in Figure represents a function, because every vertical line that intersects the graph does so exactly once.

Determine whether a graph represents a function The graph in Figure does not represent a function, because some vertical lines intersect the graph more than once.

Determine whether a graph represents a function Determine whether each graph represents a function.

Excel Graphing MS Excel Graphing functions.xls