11 X1 T02 06 relations and functions (2010)
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Transcript of 11 X1 T02 06 relations and functions (2010)
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