Detailed and comprehensive notes on functions. Includes diagrams
FEATURES
- set notation + definitons
- types of relations (one-to-one, one-to-many)
- types of graphs
- function notation
- hybrid functions
- using CAS
- graphing circles
- graphing semi circles
- inverse functions
Key words
● Set → a collection of objects (often numbers) denoted by a capital letter and { }
● Element → each object in a set is an element of that set
○ E.g. 3∈B → 3 is an element/member of set B i.e. 3 is part of set B
● Intersection (A⋂B) → the set of elements in both A and B
● Union A⋃B → the set of elements in either A or B or both
● Set difference (A/B) → the set of elements in A but not in B
● Subset (A⊆B) → reads as ‘A is is a subset of B/B is contained in A’. Means that all the
elements in A are within/included in B
● Ordered pairs → a pair of elements x and y where x is the first coordinate and y is
the second coordinate
● Relation → set of ordered pairs
● Domain → the set of all the first coordinates of the ordered pairs i.e. all the values of
x. Written left to right
● Range → the set of all the second coordinates of the ordered pairs i.e. all the values
of y. Written down to up
● Maximal/implied domain → largest domain for which the rule has meaning which is
R. If u aren’t told the domain assume it is R as that is the largest domain
Sets of Numbers
● N → set of natural number → positive integers (whole
numbers)
● Z → set of integers → positive or negative whole
numbers + zero
● Q → set of rational numbers → numbers that can be
written as a fraction or decimal
● Q’ → set of irrational numbers → numbers that can’t
be written as a fraction or have an infinite number of
digits with no recurring pattern
● R → set of all real number → any number except infinity
Set Notation
● ‘Such that’ → {x:0<x<5}
● Interval → (0,5) → reads as a set of real numbers between 0 and 5
● ⏺ → number is included
● ○ → number isn’t included
, ● [ → number is included
● ( → number isn’t included
● R⁺ → (0,∞)
● R⁻ → (-∞,0)
● R\{0} → the set of real numbers not including 0
● R → (-∞,∞)
Converting from ‘such as’ to interval notation
● {x:x>-1} → (-1,∞)
● {x:x<-2} → (-∞,-2}
How to know what is a function
● A function is a relation that passes the vertical line test → where any vertical line
can’t cross the graph more than once
● The types of relations that are functions are many-to-one or one-to-one
Types of relation
● Many-to-one → many cuts on the horizontal line and 1 cut on
the vertical line
● One-to-many → one cut on the horizontal line and many cuts
on the vertical line
● One-to-one → 1 cut on the horizontal line and 1 cuts on the
vertical line
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