Relations and Functions 2-1 Relations and Functions

Today’s objectives: Analyze and graph relations. Find functional values.

Ordered Pairs Ordered pairs are names of points in a coordinate plane. They are written in the form (x,y). They are used to describe how the set of the x-coordinates (the domain) can be related to the set of the y-coordinates (the range). Ordered pairs also determine the graph that represents a particular equation in the coordinate plane.

Cartesian Coordinate Plane y - axis Quadrant II Quadrant I origin (0,0) x - axis Quadrant III Quadrant IV

Relation Def: a set of ordered pairs The DOMAIN of a relation is the set of all x-coordinates from the ordered pairs, and the RANGE in the set of all y-coordinates. The graph of the relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation.

Function Def: a special type of relation in which each element of the domain is paired with exactly one element of the range. We can use a mapping to show how each member of the domain is paired with each member of the range. Ex: {(-3,1), (0,2), (2,4)} Domain Range -3 1 0 2 2 4

More Mapping Examples {(-3,1), (0,2), (2,4)} {(-1,5), (1,3), (4,5)} {(5,6), (-3,0), (1,1), (-3,6)} m

More Mapping Examples {(-3,1), (0,2), (2,4)} one-to-one function {(-1,5), (1,3), (4,5)} function; not one-to-one {(5,6), (-3,0), (1,1), (-3,6)} not a function m

Example 1: State the domain and range of the relation shown in the graph. Is the relation a function? (3,3) (-3,1) (1,2) (-4,0) (0,-2)

Example 2: Why is {(9,3), (9,-3), (4,2), (4, -2)} not a function?

Vertical Line Test The vertical line test can be used to determine whether or not a relation is a function by simply examining its graph. Ex:

Vertical Line Test The vertical line test can be used to determine whether or not a relation is a function by simply examining its graph. Ex: function not a function

Example 3: Graph the following data, and then tell whether or not the relation is a function. YEAR FUEL EFFICIENCY 1995 20.5 mi/gal 1996 20.8 mi/gal 1997 20.6 mi/gal 1998 20.9 mi/gal 1999 2000 2001 20.4 mi/gal

Equations and Relations Relations and functions can also be represented by equations. To find values for the relation, we make a table. Ex: For y = 2x + 1… X Y -1 1 2

Example 4: Graph the relation represented by y = 3x – 1. Find the domain and range. Determine whether or not the relation is a function.

Example 5: Graph the relation represented by x = y2 + 1. Find the domain and range. Determine whether or not the relation is a function.

Functions (continued) When an equation represents a function, the variable x (the domain) is called the INDEPENDENT VARIABLE, and y is the DEPENDENT VARIABLE because its value depends on x. Because the values of x determine the values of y, we can use FUNCTIONAL NOTATION, which more or less replaces y with f(x). Ex. y = 2x + 1 can be written as f(x) = 2x + 1. Using this notation, I can evaluate the function for a particular value of x.

Example 6: Given f(x) = x3 -3 and h(x) = 0.3x2 – 3x – 2.7, find each value. a) f(-2) b) h(1.6) c) f(2t)

HOMEWORK p. 60 #17-22, 24-34 even, 35-41