Block 3

Functions

What is to be Learned?

• What a function is• Different ways functions can be represented• Meaning of words domain and range• How to express functions

Functions are like Factories

Basic Function

Something goes in and comes out different (usually!)

In Out+ 4Domain Range

Basic Function

Something goes in and comes out different (usually!)

In Out+ 4Domain Range

Basic Function

In Out 5 -5 x

In Out+ 4

9-1

x + 4

Basic Function

Domain Range 5 -5 x

In Out+ 4

9-1

x + 4

Fancy Way of Writing

F(x) = x + 4 ↑ ↑ Domain Range F(7) ? F(7) = 7 + 4= 11

F of x

F(x) = 2x + 3 Domain { 1 , 3 , 5 }

Range?F(1)?F(1) = 2(1) + 3 = 5F(3) = 9F(5) = 13

Range { 5 , 9 , 13 }

Representing Functions

Table

X 1 2 3 4 5

F(x) 5 7 9 11 13

F(x) = 2x + 3

Graph

x

F(x)

just like Y

f(x) = 2x + 3

Daft looking things

Domain Range

5 13

10 2311 2519 41

f(x) = 2x + 3

Definition

For every value of x in the domain, there is exactly one value of f (or y) in the range

Domain Range

f(x) = 5x + 9 f(x + 3)?

f(x + 3) =

f(x) = 5x + 9 f(x + 3)?

f(x + 3) =

f(x) = 5x + 9 f(x + 3)?

f(x + 3) =

f(x) = 5x + 9 f(x + 3)?

f(x + 3) =

f(x) = 5x + 9 f(x + 3)?

f(x + 3) =

= 5x + 15 + 9 = 5x + 24

5 + 9x + 3( )x

f(x) = 3x – 2

f(x + 5) = 3 = 3x + 15 – 2 = 3x + 13

– 2 x + 5( )

f(x + 5)?

f(x) = x2

f(x + 3) =

= (x + 3)(x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9

2x + 3( )

f(x + 3)?

Functions

In functions the domain values (usually x) give one range value (usually f(x) or y)

Can be represented by formulae, graphs, tables or daft diagrams.

f(x) = x2 + 6f(4) =

f(x + 3) = 42 + 6(x + 3)2 + 6

g(x) = x2 + 4x – 3g(5) =

g(x – 7 ) = 52 + 4(5) – 3(x – 7)2 + 4(x – 7) – 3

f(x) = 6x – 2

f(x + 4) = 6 = 6x + 24 – 2 = 6x + 22

– 2 x + 4( )

f(x + 4)?Key Question

Forbidden Domain Values!

Sums that do not work!!!!!

• Dividing by zero• Square root of a negative

Applying to Functions

f(x) = 10/x

Domain: x cannot equal zero

x = 0

f(x) = √xDomain: x must not be negative

x ≥ 0

Sneakier

f(x) = 10/x-3

Domain: x = 3

f(x) = √x – 5 Domain x ≥ 5

For some functions there are domain values which are undefined

Ex f(x) = √x – 7Domain x ≥ 7