Chapter 2 Section 1 Relations and Functions

ALGEBRA 2 LESSON 2-1 Graph each ordered pair on the coordinate plane. 1. (–4, –8) 2. (3, 6) 3. (0, 0) 4. (–1, 3) 5. (–6, 5) Evaluate each expression for x = –1, 0, 2, and x –2x x |x – 3| (For help, go to Skills Handbook page 848 and Lesson 1-2.)

Solutions x + 2 for x = –1, 0, 2, and 5: –1 + 2 = 1; = 2; = 4; = 7 7.–2x + 3 for x = –1, 0, 2, and 5: –2(–1) + 3 = = 5; –2(0) + 3 = = 3; –2(2) + 3 = –4 + 3 = –1; –2(5) + 3 = – = –7 8.2x for x = –1, 0, 2, and 5: 2 (–1) = = = 3; = = = 1; = = = 9; = = = 51 9.|x – 3| for x = –1, 0, 2, and 5: |–1 – 3| = |–4| = 4; |0 – 3| = |–3| = 3; |2 – 3| = |–1| = 1; |5 – 3| = |2| = 2

Relations Relation – a set of pairs of input and output values  Can be written in ordered pairs (x,y)  Can be graphed on a coordinate plane Domain – the set of all input values  The x values of the ordered pairs Range – the set of all output values  The y values of the ordered pairs When writing domains and ranges:  Use braces { }  Do not repeat values NON-Example: {3,3,5,7,9}

Graph the relation {(–3, 3), (2, 2), (–2, –2), (0, 4), (1, –2)}.

Try These Problems a){(0,4),(-2,3),(-1,3),(-2,2),(1,-3)}b){(-2,1),(-1,0),(0,1),(1,2)}

Write the ordered pairs for the relation. Find the domain and range. {(–4, 4), (–3, –2), (–2, 4), (2, –4), (3, 2)} The domain is {–4, –3, –2, 2, 3}. The range is {–4, –2, 2, 4}.

Mapping Diagram Another way to represent a relation (beside traditional graphing) Links elements of the domain with corresponding elements of the range How To make a mapping diagram:  Make two lists – place numbers from least to greatest Domains on the left Ranges on the Right  Draw arrows from corresponding domains to ranges (x’s to y’s)

Make a mapping diagram for the relation {(–1, 7), (1, 3),(1, 7), (–1, 3)}. DomainRange

Try These Problems Make a Mapping Diagram for each relation. a){(0,2),(1,3),(2,4)}b){(2,8),(-1,5),(0,8),(-1,3),(-2,3)}

Functions Function – a relation in which each element of the domain is paired with EXACTLY one element of the range All functions are relations, but not all relations are functions!!!

There are several ways to determine if a relation is a function: Mapping Diagram  If any element of the domain (left) has more than one arrow from it List of ordered pairs  Look to see if any x values are repeated Coordinate Plane  Vertical Line Test – If a vertical line passes through more than one point on the graph then the relation is NOT a function.

Determine whether each relation is a function a) b) This is NOT a function because -2 is paired with both -1 and 3. This is a function because every element of the domain is paired with exactly one element of the range.

Try These Problems Determine whether each relation is a function a) b) Function Not a function

Vertical Line Test

Try These Problems Use the Vertical Line Test to determine whether each graph represents a function. a) b)c) Function Not a Function

Function Rules Function Rule – expresses an output value in terms of an input value  Examples: y = 2x f(x) = x + 5 C = πd Function Notation –  f(x) is read as “f of x”  This does NOT mean f times x !!!!  f(3) is read as “f of 3”: It means evaluate the function when x = 3. (plug 3 into the equation)  Any letters may be used C(d) h(t)

Find ƒ(2) for each function. a.ƒ(x) = –x ƒ(2) = – = –4 + 1 = –3 b.ƒ(x) = |3x| ƒ(2) = |3 2| = |6| = 6 c.ƒ(x) = 9 1 – x ƒ(2) = = = –9 9 1 – 2 9 –1

Try These Problems Find f(-3), f(0), and f(5) for each function a)f(x) = 3x – 5 f(-3) = 3(-3) – 5 = -9 – 5 = -14 f(0) = 3(0) – 5 = 0 – 5 = -5 f(5) = 3(5) – 5 = 15 – 5 = 10 a)f(a) = ¾ a – 1 f(-3) = ¾ (-3) – 1= -9 / 4 – 4 / 4 = - 13 / 4 f(0) = ¾ (0) – 1 = 0 – 1 = -1 f(5) = ¾ (5) – 1 = 15 / 4 – 4 / 4 = 11 / 4 a)f(y) = - 1 / 5 y + 3 / 5 f(-3) = - 1 / 5 (-3) + 3 / 5 = 3 / / 5 = 6 / 5 f(0) = - 1 / 5 (0) + 3 / 5 = / 5 = 3 / 5 f(5) = - 1 / 5 (5) + 3 / 5 = - 5 / / 5 = -2 / 5

Homework Day 1:  Practice 2.1 All  Show all work for function notation problems (you may need another sheet of paper) Day 2:  Page 59 # 1 – 21 odd, all  Be sure to write each problem – this includes writing out the sets of coordinate pairs and sketching graphs  Show all work for function notation problems