CIE A Level Maths: Pure 1复习笔记1.3.1 Language of Functions

Language of functions

• The language of functions has many keywords associated with it that need to be understood

What are mappings?

• A mapping takes an ‘input’ from one set of values to an ‘output’ in another

• Mappings can be
  • ‘many-to-one’ (many ‘input’ values go to one ‘output’ value)
  • ‘one-to-many’
  • ‘many-to-many’
  • ‘one-to-one’

What is the difference between a mapping and a function?

• A function is a mapping where every ‘input’ value maps to a single ‘output’
• Many-to-one and one-to-one mappings are functions
• Mappings which have many possible outputs are not functions

Notation

• Functions are denoted by the notation f(x), g(x), etc
  • eg. f(x) = x² - 3x + 2
• Of the alternative notation
  • eg. f : x ↦ x² – 3x + 2

Sets of numbers

• Functions often involve domains and ranges for specific sets of numbers
• All numbers can be organised into different sets ℕ, ℤ, ℚ, ℝ

• So ℕ is a subset of ℤ etc
• ℤ^- would be the set of negative integers only

Domain

• The domain of a function is the set of values that are allowed to be the ‘input’

• A function is only fully defined once its domain has been stated
• Restrictions on a domain can turn many-to-one functions into one-to-one functions

Range

• The range of a function is the set of values of all possible ‘outputs’
• The type of values in the range depend on the domain