Thursday
Thursday
Relations and Functions Chapter 2 Section 2-1 Pages 72-81
Objectives • I can find Domain and Range • I can determine if the relation is a function by two methods • IMPORTANT VOCABULARY in this section!!
Important Vocabulary • Relation • Domain • Range • Function • Vertical Line Test
Cartesian Coordinate Plot • x-axis: This is the horizontal axis. • y-axis: This is the vertical axis • Origin: This is the center point (0,0) • Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis.
Quadrants • The coordinate system is divided into 4 quadrants. • Look at the overhead slide 2-1.1 to see how this entire system works.
y-axis 9 8 7 6 5 4 3 2 10 -9 -8 -7 -6 -2 1 0 -5 -4 -3 -1 x-axis 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 Quadrant I (+, +) Quadrant II (-, +) Origin (0,0) Quadrant IV (+, -) Quadrant III (-, -)
Relation • A relation is a set of ordered pairs! • Need the { } to show a set • Example: { (1, 2), (3, 4), (5, 6) }
Domain and Range • The domain in any relation is the first coordinates from the ordered pairs. It is the Input! • Domain = X -Values • The range in any relation is the second coordinates from the ordered pairs. It is the Output! • Range = Y- Values
Dependent Variable
Example 1: Domain/Range • Given the following relation • {(2,3), (-4,8), (2,6), (7,-3)} • What is the Domain? • { -4, 2, 7} • **Notice they are listed least to greatest!! • No duplicates!!! • What is the Range? • {-3, 3, 6, 8}
Example 2: • Given the following ordered pairs, find the domain and range. • {(4,5), (-2,3), (5,6)} • Domain is {-2, 4, 5} • Range is {3, 5, 6}
Answer Format • When listing a set of numbers for domain or range, use the set symbols {} • List numbers from least to greatest (increasing order). No duplicates! • Ex: the domain has numbers: 3, -2, 5, 2, 3 • {-2, 2, 3, 5}
for Example 1 GUIDED PRACTICE 1. Identify the domain and range of the function. SOLUTION The domain is the set of inputs:{0,1,2,4}The range is the set of outputs:{1,2,5}
4 Ways to see Relations
Relations Ordered Pairs (2, 3) (-3, 1) (1, -2) Tables Graphs X Mapping Y 2 -3 1 3 1 -2
Function • A function is a special relation in which • NO DUPLICATED “x-values” • Example: Is the following relation a function: { (1,3), (4,-9), (6,3) } • Answer: Yes. Each element from the domain has only 1 element from the range
Ex 2: How about this relation. Is it a function? • Given the following { (2,3), (-4,8), (2,6), (7,-3)} • Domain? • { -4, 2, 7} • Range? • {-3, 3, 6, 8} • Function: No. the element 2 from the domain is mapped to 2 elements in the range.
EXAMPLE 2 Identify a function Tell whether the pairing is a function. a. The pairing is not a function because the input 0ispaired with both 2 and 3.
Output Input 0 0 1 2 8 4 6 12 EXAMPLE 2 Identify a function b. The pairing is a function because each input is pairedwith exactly one output.
36912 1221 2. Input 3 6 9 12 Output 1 2 2 1 for Example 2 GUIDED PRACTICE Tell whether the pairing is a function. SOLUTION The pairing is a function because each input is pairedwith exactly one output.
3. Input 2 2 4 7 Output 0 1 2 3 247 0123 for Example 2 GUIDED PRACTICE Tell whether the pairing is a function. SOLUTION The pairing is not a function because each input is notpaired with exactly one output.
Vertical Line Test You can use a vertical line test to easily see on a graph is the relation is a function. You place a straight edge like a pencil vertical on the graph and move it across the graph. If one line intersects the graph at only one point, then it is a function.
Applying VLT
y y y x2 + y2 = 1 y = x2 y2= x x x x Vertical Line Test Vertical Line Test Consider the graphs. 2 points of intersection 1 point of intersection 2 points of intersection
Examples VLT
VLT Applies to Grid
½ Sheet Activity • On your ½ sheet of paper • Make 4 relations that we discussed today. • ONE of each type • Trade with your partner and detrmine • Domain and Range • Is each a function?
Homework • WS 1-5
