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VCE Maths Methods - Functions & Relations
Functions and relations
• Relations• Set rules & symbols• Sets of numbers• Sets & intervals• Functions• Relations• Function notation• Hybrid functions• Hyperbola• Truncus• Square root• Circle• Inverse functions
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VCE Maths Methods - Functions & Relations
Relations
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• A relation is a rule that links two sets of numbers: the domain & range.• The domain of a relation is the set of the !rst elements of the ordered
pairs (x values).• The range of a relation is the set of the second elements of the ordered
pairs (y values).• (The range is a subset of the co-domain of the function.)• Some relations exist for all possible values of x.• Other relations have an implied domain, as the function is only valid for
certain values of x.
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VCE Maths Methods - Functions & Relations
Set rules & symbols
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• ∈ : element of a set.
• ∉ : not an element of a set.
• ∩ : intersection of two sets (in both B and C)
• ∪ : union of two sets (elements in set B or C)
• ⊂ : subset of a set (all elements in D are part of set B)
• \ : exclusion (elements in set B but not in set C)
• Ø : empty set
B = {2, 4, 6, 8, 10}(even numbers)
A = {1, 3, 5, 7, 9}(odd numbers)
C = {2, 3, 5, 7}(prime numbers)
D = {4,8}(multiples of 4)
3! A
6! A
B!C = {2}
B!C = {2,3,4,5,6,7,8,10}
D !B
B \ C = {2,6,10}
A!B ="
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VCE Maths Methods - Functions & Relations
RQ
Sets of numbers
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• The domain & range of a function are each a subset of a particular larger set of numbers.
• Natural numbers (N): {1, 2, 3,4, ........}
• Integers (Z): {-2, -1, 0, 1, 2, ........}
• Rational numbers (Q): Any numbers that can be made from the division of two integers (but not dividing by 0) eg 1/3, - 2.45, 5.787878.....
• Real numbers (R): The set of all rational and irrational numbers (includes surds, ", e)
ZNN is a subset of Z
N ! Z N ! Z !Q !R
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VCE Maths Methods - Functions & Relations
Sets & intervals
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• Intervals of the real numbers can be depicted using the appropriate brackets & set notations.
• Square brackets include point, round brackets don’t.
{x : 0< x <3}
!",0( )
!4,2( ]
0,3( )
{x :!4< x "2}
{x : x <0}
{x : x <!2}"{x : x #1}
R!
Set Interval
!",!2( )# 1,"[ ) R \ !2,1[ )
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VCE Maths Methods - Functions & Relations
Functions
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• A function is a type of relation where each value of x has a unique value of y.
• Functions can be one to one or many to one - where more than one x value shares the same y value.
• Examples include linear functions and circular functions (sin x, cos x)
• Functions can be determined by the vertical line test - any vertical line will cut through a function only once.
Linear function: one to one Circular function: many to one
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VCE Maths Methods - Functions & Relations
Relations
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• A relation can also be one to many or many to many - where x values can have more than one y value.
• A circle is an example of this of a many to many function.
• A vertical line can cut through this graph more than once.
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VCE Maths Methods - Functions & Relations
Function notation
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• Function notation is used to describe the domain & any restrictions that might be in place.
f :[0,!)"R ,f (x )= x 2
Domain (restricted)
The name of the function
Co-domain
Rule
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VCE Maths Methods - Functions & Relations
Hybrid functions
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Flat rate: $100 for the !rst 400 km10c for each km between
400 km & 800 km
20c for each km after 800 km
C(x) = 100 for 0<x !400
C(x) = 0.2(x-800)+140 for 800< x
Cost ($)
Distance (km)
C(x) = 0.1(x-400)+100 for 400< x !800
• A hybrid (piecewise) function is one where di#erent rules apply over di#erent parts of the domain.
• For example, the cost of car hire di#ers according to the distance traveled.
C(x) = 100, 0! x !8000.1(x-400)+100, 400<x !8000.2(x-800)+140, 800<x
"
#$
%$
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VCE Maths Methods - Functions & Relations
Hyperbola
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y =1
x
y = 1
(x !h)+k
y = 1
(x !4)!1
Range: R\{-1}
Domain: R\{4}
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VCE Maths Methods - Functions & Relations
Truncus
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y = 1
x 2
y = 1
(x !h)2 +k
y = 1
(x+2)2 +3
Range: (3,$)
Domain: R\{-2}
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VCE Maths Methods - Functions & Relations
Square root
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y = x
y = x !h +k
y = x+3+2
Range: [2,$)
Domain: [-3, $)
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VCE Maths Methods - Functions & Relations
Circle
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x2+ y 2 = r 2
x2+ y 2 =36
(x !h)2+( y !k )2 = r 2
(x !2)2+( y +4)2 =36
Range: [-10,2]
Domain: [-4,8]
(Diameter = 12)
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VCE Maths Methods - Functions & Relations
• Circles are described by a relation, not a function.
• They can be de!ned by combining two functions together.
Circle from functions
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x2+ y 2 =36
y2 =36! x 2
y = ± 36! x 2
y = 36!x 2
y =! 36!x 2
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VCE Maths Methods - Functions & Relations
Inverse functions
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• Every one to one function has an inverse function, f-1(x).
• Inverses are used to work backwards & solve equations.
• The graph of an inverse function can be found from mirroring the graph around the line y = x.
• The domain of the inverse f-1(x) is the range of f(x).
• The range of the inverse f-1(x) is the domain of f(x).
• The x intercept of f-1(x) is the y intercept of f(x).
• The y intercept of f-1(x) is the x intercept of f(x).
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VCE Maths Methods - Functions & Relations
Inverse functions - from graphs
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y intercept = 1
x intercept = 1Asymptote: y = 0
Asymptote: x = 0
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VCE Maths Methods - Functions & Relations
Inverse functions - from equations
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y = x
2!3
x intercept = 6y intercept = -3
y intercept = 6
x intercept = -3
y =2x+6 y
x
f (x ) : y =2x+6
f!1(x ) : x =2y +6
2y = x !6
y = x
2!3