11 X1 T02 06 relations and functions (2010)

Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2  y 2  25

e.g. x 2  y 2  25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y  x 2

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
y 1
y
x

x

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
y 1
y
x

x
function

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
y 1 y x  y2
y
x

x x
function

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
y 1 y x  y2
y
x
function
x x
function

e.g. x 2  y 2  25
of one y value.
e.g. y  x 2
Straight Line Test
y 1 y x  y2
y
x
function
x x
 note: actually two functions 
function  
 y  x and y   x 

Domain and Range y  f  x 

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

function/relation.
“Domain is the INPUT of the function/relation”

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

function/relation.
e.g.
y x  y2

x

function/relation.
e.g.
y x  y2

x

domain: x  0

function/relation.
e.g.
y x  y2 y y  f  x
3
1
x 2 x

domain: x  0

function/relation.
e.g.
y x  y2 y y  f  x
3
1
x 2 x

domain: x  0 domain: x  0 and x  2

Things to look for:
1. Fractions:

Things to look for:
1. Fractions: bottom of fraction  0

Things to look for:
1
e.g.  i  y 
x

Things to look for:
1
e.g.  i  y 
x
x0

Things to look for:
1
e.g.  i  y 
x
x0
domain: all real x except x  0

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1
domain: all real x except x  1

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1
4x 3
 iii  y  
x 1 7  x

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1
4x 3
 iii  y  
x 1 7  x

x 1  0
x 1

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1
4x 3
 iii  y  
x 1 7  x

x 1  0 7x  0
x 1 x7

Things to look for:
1 1
e.g.  i  y   ii  y 
x x2 1
x0 x2 1  0
x2  1
x  1
4x 3
 iii  y  
x 1 7  x

x 1  0 7x  0
x 1 x7
domain: all real x except x  1 or 7

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


e.g.  i  y  4  x 2


e.g.  i  y  4  x 2
4  x2  0
x2  4


e.g.  i  y  4  x 2
4  x2  0
x2  4
domain:  2  x  2


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0
x2  4
domain:  2  x  2


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0
x2  4 x  3
domain:  2  x  2


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5

1
 iii  y 
x2


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5

1
 iii  y 
x2

x20


e.g.  i  y  4  x 2  ii  y  x  3  5  x
4  x2  0 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5

1
 iii  y 
x2

x20
domain: x  2

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

“Range is the OUTPUT of the function/relation”


e.g.
y x  y2

x


e.g.
y x  y2

x

range: all real y


e.g.
y x  y2 y y  f  x
3
1
x 2 x

range: all real y


e.g.
y x  y2 y y  f  x
3
1
x 2 x

range: all real y range: y  1 and y  3

Things to look for:
1. Maximum/Minimum values:

Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always  0

Things to look for:
are always  0
e.g.  i  y  x 2

Things to look for:
are always  0
e.g.  i  y  x 2
range: y  0

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2
y  50

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2
y  50
range: y  5

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2  iv  y  x  2
y  50
range: y  5

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5
v y  x  2  5

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5
v y  x  2  5
y  05

Things to look for:
are always  0
e.g.  i  y  x 2  ii  y  x 2  3
range: y  0 y  03
range: y  3

 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5
v y  x  2  5
y  05
range: y  5

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


e.g. y  4  x 2


e.g. y  4  x 2
domain:  2  x  2


e.g. y  4  x 2 when x  2, y  4  22
domain:  2  x  2 0


e.g. y  4  x 2
when x  2, y  4  22 when x  0, y  4  02
domain:  2  x  2 0 2

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

3. Fractions:

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

3. Fractions: If you have a constant on the top of the fraction, fraction  0

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
x

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
x
y0

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
x
y0
range: all real y except y  0

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50
range: all real y except y  0 range: all real y except y  5

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50

x7
 iii  y 
x4

when x  2, y  4  22 when x  0, y  4  0
2
e.g. y  4  x 2

domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50

x7 1
 iii  y  x4 x7
x4
x4
3


e.g. y  4  x 2
when x  2, y  4  22 when x  0, y  4  02
domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50

x7 1
 iii  y  x4 x7
x4
3 x4
y  1 3
x4


e.g. y  4  x 2
when x  2, y  4  22 when x  0, y  4  02
domain:  2  x  2 0 2
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y0 y  50

x7 1
 iii  y  x4 x7
x4
3 x4
y  1 3
x4
y  1 0

Function Notation
e.g. f  x   3 x 2  4

Function Notation
e.g. f  x   3 x 2  4
a) f  5

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4
2

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4
2

 75  4
 79

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a 
2

 75  4
 79

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a   3a 2  4
2

 75  4
 79

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a   3a 2  4
2

 75  4
 79
c) f  x  h   f  x 

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a   3a 2  4
2

 75  4
 79
c) f  x  h   f  x   3  x  h   4   3x 2  4 
2

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a   3a 2  4
2

 75  4
 79
c) f  x  h   f  x   3  x  h   4   3x 2  4 
2

 3 x 2  6 xh  3h 2  4  3 x 2  4
 6 xh  3h 2

Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a   3a 2  4
2

 75  4
 79
c) f  x  h   f  x   3  x  h   4   3x 2  4 
2

 3 x 2  6 xh  3h 2  4  3 x 2  4
 6 xh  3h 2

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

11 X1 T02 06 relations and functions (2010)

More Related Content

What's hot

Viewers also liked

Similar to 11 X1 T02 06 relations and functions (2010)

More from Nigel Simmons

Recently uploaded

11 X1 T02 06 relations and functions (2010)