Chapter 2 Sets, Relations, and Functions. 2 2.1 SET OPERATIONS The union of A and B.
2.1 – Relations & Functions
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2.1 – Relations & Functions
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2.1 – Relations & Functions. 2.1 – Relations & Functions. Relation. 2.1 – Relations & Functions. Relation – a set of ordered pairs. 2.1 – Relations & Functions. Relation – a set of ordered pairs (a relationship between numbers). 2.1 – Relations & Functions. - PowerPoint PPT Presentation
Transcript of 2.1 – Relations & Functions
2.1 – Relations & Functions
2.1 – Relations & FunctionsRelation
2.1 – Relations & FunctionsRelation – a set of ordered pairs
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function – relation where each x has only one y value
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function – relation where each x has only one y value
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function – relation where each x has only one y value
Note:
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function – relation where each x has only one y value
Note: each y can have more than one x value!
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)
ordered pairs – (x, y)
domain range
Function – relation where each x has only one y value
Note: each y can have more than one x value!
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)ordered pairs – (x, y)
domain rangeFunction – relation where each x has only one
y valueNote: each y can have more than one x value!
*If each y does have only one x value, it is called a one-to-one function.
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)ordered pairs – (x, y)
domain rangeFunction – relation where each x has only one
y valueNote: each y can have more than one x value!
*If each y does have only one x value, it is called a one-to-one function.
2.1 – Relations & FunctionsRelation – a set of ordered pairs
(a relationship between numbers)ordered pairs – (x, y)
domain rangeFunction – relation where each x has only one
y valueNote: each y can have more than one x value!
*If each y does have only one x value, it is called a one-to-one function.
Example 1
Example 1 {(-3,1),(0,2),(2,4)}
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
0
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
0
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
0
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3
0
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2 4
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2 4
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2 4
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1
0 2
2 4
Example 1 {(-3,1),(0,2),(2,4)}Domain Range
-3 1 0 2 2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}
Example 1 {(-3,1),(0,2),(2,4)}Domain Range
-3 1 0 2 2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}Domain Range Domain Range
Example 1 {(-3,1),(0,2),(2,4)}Domain Range
-3 1 0 2 2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}Domain Range Domain Range
-1 -3 0 1 3 1 1 4 5 5 6
Example 1 {(-3,1),(0,2),(2,4)}Domain Range
-3 1 0 2 2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}Domain Range Domain Range
-1 -3 0 1 3 1 1 4 5 5 6
Example 1 {(-3,1),(0,2),(2,4)}Domain Range
-3 1 FUNCTION
0 2 1-1 2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}Domain Range Domain Range
-1 -3 0 1 3 1 1 4 5 5 6
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1 FUNCTION
0 2 1-1
2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}
Domain Range Domain Range
-1 -3 0
1 3 1 1
4 5 5 6
FUNCTION
Example 1 {(-3,1),(0,2),(2,4)}
Domain Range
-3 1 FUNCTION
0 2 1-1
2 4
{(-1,5),(1,3),(4,5)} {(5,6),(-3,0),(1,1),(-3,6)}
Domain Range Domain Range
-1 -3 0
1 3 1 1
4 5 5 6
FUNCTION Not a Function
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1),(0,1),(-1,1)}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain:
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2,
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1,
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Range:
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Range: {-1,
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Range: {-1, 1}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Range: {-1, 1}
Example 2 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.
(a)
{(-2,1),(-1,-1,),(0,1),(-1,1)}
Domain: {-2, -1, 0}
Range: {-1, 1}
Not a Function!!!
(b) y = 3x
Domain:
Range:
(b) y = 3x
Domain:
Range:
x y
(b) y = 3x
Domain:
Range:
x y
-1
0
1
(b) y = 3x
Domain:
Range:
x y
-1 -3
0 0
1 3
(b) y = 3x
Domain:
Range:
x y
-1 -3
0 0
1 3
(b) y = 3x
Domain:
Range:
x y
-1 -3
0 0
1 3
(b) y = 3x
Domain: all real numbers
Range:
x y
-1 -3
0 0
1 3
(b) y = 3x
Domain: all real numbers
Range: all real numbers
x y
-1 -3
0 0
1 3
(b) y = 3x
Domain: all real numbers
Range: all real numbers
x y
-1 -3
0 0
1 3
1-1 FUNCTION
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) =
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z) = (2z)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z) = (2z)
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z) = (2z)
Example 3Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3) f(x) = 3x – 5 f(-3) = 3(-3) – 5
= -9 – 5 = -14(b) g(2z)
g(x) = x2 + 2 g(2z) = (2z)2
Example 3Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3) f(x) = 3x – 5 f(-3) = 3(-3) – 5
= -9 – 5 = -14(b) g(2z)
g(x) = x2 + 2 g(2z) = (2z)2
Example 3Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3) f(x) = 3x – 5 f(-3) = 3(-3) – 5
= -9 – 5 = -14(b) g(2z)
g(x) = x2 + 2 g(2z) = (2z)2 + 2
Example 3Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3) f(x) = 3x – 5 f(-3) = 3(-3) – 5
= -9 – 5 = -14(b) g(2z)
g(x) = x2 + 2 g(2z) = (2z)2 + 2
= (2)2
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z) = (2z)2 + 2
= (2)2(z)2
Example 3
Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3)
f(x) = 3x – 5
f(-3) = 3(-3) – 5
= -9 – 5 = -14
(b) g(2z)
g(x) = x2 + 2
g(2z) = (2z)2 + 2
= (2)2(z)2 + 2
Example 3Given f(x) = 3x – 5 and g(x) = x2 + 2, find:
(a) f(-3) f(x) = 3x – 5 f(-3) = 3(-3) – 5
= -9 – 5 = -14(b) g(2z)
g(x) = x2 + 2 g(2z) = (2z)2 + 2
= (2)2(z)2 + 2
= 4z2 + 2
