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