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Relations and Functions
2-1 Relations and Functions
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Today’s objectives: Analyze and graph relations.
Find functional values.
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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.
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Cartesian Coordinate Plane
y - axis Quadrant II Quadrant I origin (0,0) x - axis Quadrant III Quadrant IV
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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.
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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
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More Mapping Examples {(-3,1), (0,2), (2,4)} {(-1,5), (1,3), (4,5)}
{(5,6), (-3,0), (1,1), (-3,6)} m
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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
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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)
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Example 2: Why is {(9,3), (9,-3), (4,2), (4, -2)} not a function?
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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:
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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
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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
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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
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Example 4: Graph the relation represented by y = 3x – 1.
Find the domain and range. Determine whether or not the relation is a function.
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Example 5: Graph the relation represented by x = y2 + 1.
Find the domain and range. Determine whether or not the relation is a function.
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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.
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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)
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HOMEWORK p. 60 #17-22, even, 35-41
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