Relations & Functions

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

function

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

function

y

x

2x y

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

function

y

x

2x y

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

function

y

x

2x y

function

Relations & Functions2 2e.g. 25x y

A relation is a set of any ordered pairs that are related in any way.

2e.g. y x

A function is a relation such that for any x value, there is a maximum of one y value.

Straight Line TestIf a straight line is drawn parallel to the y axis, it will only cross a function once, if at all.

y

x

1yx

function

y

x

2x y

function

note: actually two functions

and y x y x

Domain and Range y f x

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.“Domain is the INPUT of the function/relation”

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

To find a domain, look for values x could not be.

“Domain is the INPUT of the function/relation”

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

To find a domain, look for values x could not be.e.g.

y

x

2x y

“Domain is the INPUT of the function/relation”

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

To find a domain, look for values x could not be.e.g.

y

x

2x y

domain: 0x

“Domain is the INPUT of the function/relation”

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

To find a domain, look for values x could not be.e.g.

y

x

2x y

domain: 0x

y

x213

y f x

“Domain is the INPUT of the function/relation”

Domain and Range y f x

Domain: All possible values of x that can be substituted into the function/relation.

To find a domain, look for values x could not be.e.g.

y

x

2x y

domain: 0x

y

x213

y f x

domain: 0 and 2x x

“Domain is the INPUT of the function/relation”

Things to look for:

1. Fractions:

Things to look for:

1. Fractions: bottom of fraction 0

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

2 11

xx

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

domain: all real except 1x x

2 11

xx

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

domain: all real except 1x x

2 11

xx

4 3 1 7

xiii yx x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

domain: all real except 1x x

2 11

xx

4 3 1 7

xiii yx x

1 0x

1x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

domain: all real except 1x x

2 11

xx

4 3 1 7

xiii yx x

1 0x

1x

7 0x

7x

Things to look for:

1. Fractions: bottom of fraction 0

1e.g. i yx

0x domain: all real except 0x x

21

1ii y

x

2 1 0x

domain: all real except 1x x

2 11

xx

4 3 1 7

xiii yx x

1 0x

domain: all real except 1 or 7x x

1x

7 0x

7x

2. Root Signs:

2. Root Signs: you can’t find the square root of a negative number.

2. Root Signs:

2e.g. 4i y x

you can’t find the square root of a negative number.

2. Root Signs:

2e.g. 4i y x 24 0x

you can’t find the square root of a negative number.

2 4x

2. Root Signs:

2e.g. 4i y x 24 0x

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

5 05

xx

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

domain: 3 5x

5 05

xx

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

domain: 3 5x

5 05

xx

1 2

iii yx

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

domain: 3 5x

5 05

xx

1 2

iii yx

2 0x

you can’t find the square root of a negative number.

2 4x domain: 2 2x

2. Root Signs:

2e.g. 4i y x 24 0x

3 5ii y x x

3 03

xx

domain: 3 5x

5 05

xx

1 2

iii yx

2 0x domain: 2x

you can’t find the square root of a negative number.

2 4x domain: 2 2x

Range: All possible y values obtained by substituting in the domain

Range: All possible y values obtained by substituting in the domain“Range is the OUTPUT of the function/relation”

Range: All possible y values obtained by substituting in the domain

e.g.y

x

2x y

“Range is the OUTPUT of the function/relation”

Range: All possible y values obtained by substituting in the domain

e.g.y

x

2x y

range: all real y

“Range is the OUTPUT of the function/relation”

Range: All possible y values obtained by substituting in the domain

e.g.y

x

2x y

range: all real y

y

x213

y f x

“Range is the OUTPUT of the function/relation”

Range: All possible y values obtained by substituting in the domain

e.g.y

x

2x y

range: all real y

y

x213

y f x

range: 1 and 3y y

“Range is the OUTPUT of the function/relation”

Things to look for:

1. Maximum/Minimum values:

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y x

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

2iv y x

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

2iv y x range: 0y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

2iv y x range: 0y

2 5v y x

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

2iv y x range: 0y

2 5v y x 0 5y

Things to look for:

1. Maximum/Minimum values: even powers and absolute valuesare always 0

2e.g. i y xrange: 0y

2 3ii y x 0 3y

range: 3y

2 5iii y x

5 0y range: 5y

2iv y x range: 0y

2 5v y x 0 5y

range: 5y

2. Restrictions on Domain:

2. Restrictions on Domain: sub in endpoints and centre of domain

2. Restrictions on Domain:2e.g. 4y x

sub in endpoints and centre of domain

2. Restrictions on Domain:2e.g. 4y x

sub in endpoints and centre of domain

domain: 2 2x

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

sub in endpoints and centre of domain

domain: 2 2x

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

sub in endpoints and centre of domain

domain: 2 2x

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

sub in endpoints and centre of domain

domain: 2 2x

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

7 4

xiii yx

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

7 4

xiii yx

4 7x x

4x

1

3

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

7 4

xiii yx

314

yx

4 7x x 4x

1

3

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

7 4

xiii yx

314

yx

4 7x x 4x

1

3

1 0y

2. Restrictions on Domain:2e.g. 4y x 2when 2, 4 2

0x y

2when 0, 4 0 2

x y

range: 0 2y

If you have a constant on the top of the fraction, fraction 0

sub in endpoints and centre of domain

domain: 2 2x

3. Fractions:

1e.g. i yx

0y range: all real except 0y y

1 5ii yx

5 0y range: all real except 5y y

7 4

xiii yx

314

yx

4 7x x 4x

1

3

1 0y

range: all real except 1y y

Function Notation

Function Notation

2e.g. 3 4f x x

) 5a f

Function Notation

2e.g. 3 4f x x

) 5a f

Function Notation

2e.g. 3 4f x x

23 5 4

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a 23 4a

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a 23 4a

) c f x h f x

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a 23 4a

2 23 4 3 4x h x ) c f x h f x

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a 23 4a

2 23 4 3 4x h x ) c f x h f x 2 2 23 6 3 4 3 4x xh h x

26 3xh h

) 5a f

Function Notation

2e.g. 3 4f x x

75 479

23 5 4 ) b f a 23 4a

2 23 4 3 4x h x ) c f x h f x 2 2 23 6 3 4 3 4x xh h x

Exercise 2F; 1, 2, 3acdfi, 4begh, 5a, 6, 7a, 8abd,10abdf, 11aceh, 12bd, 14*

26 3xh h