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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
Relations
- A relation is a rule that links two sets of numbers: the domain & range.
- The domain of a relation is the set of the fi 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.
Q^ R
Sets of numbers
- 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, π , e) N Z N is a subset of Z N! Z N! Z! Q! R
Sets & intervals
- 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 )
Relations
- 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.
Function notation
- 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
Hyperbola
y =
x y =
(x! h)
(x! 4 )
Range: R{-1} Domain: R{4}
Truncus
y =
x 2 y =
(x! h) 2
(x + 2 ) 2
Range: (3, ∞ ) Domain: R{-2}
Circle
x 2
- y 2 = r 2 x 2
- 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)
- Circles are described by a relation, not a function.
- They can be de fi ned by combining two functions together.
Circle from functions
x 2
- y 2 = 36 y 2 = 36! x 2 y = ± 36! x 2 y = 36! x 2 y =! 36! x 2
Inverse functions - from graphs
y intercept = 1 Asymptote: y = 0 x intercept = 1 Asymptote: x = 0
Inverse functions - from equations
y = x 2 ! 3 x intercept = 6 y intercept = - y intercept = 6 x intercept = - y y = 2 x + 6 x f (x ) : y = 2 x + 6 f ! 1 (x ) : x = 2 y + 6 2 y = x! 6 y = x 2 ! 3
