11 x1 t02 06 relations & functions (2013)

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  x2
 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  x2
 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  x2
 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  x2
 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  x2
 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  x2
 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  x2
 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  x2
 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


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


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

3. Fractions:


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

3. Fractions: If you have a constant on the top of the fraction, fraction  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
range: 0  y  2

1
e.g.  i  y 
x


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 
x
y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
range: 0  y  2

1
e.g.  i  y 
x
y0
range: all real y except y  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
range: 0  y  2

1
e.g.  i  y 
1  ii  y  5 
x x
y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
range: 0  y  2

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


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
 iii  y 
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
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 & functions (2013)

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