Edexcel IGCSE Maths 复习笔记 3.2.4 Functions - Domain, Range & Exclusions

Edexcel IGCSE Maths 复习笔记 3.2.4 Functions - Domain, Range & Exclusions

What are functions?

  • Functions are a formal way of writing mathematical expressions
    • eg. f(x) = 3x + 2 would be a linear function
    • eg2. g : x ⟼ x2 + 3x – 5 would be a quadratic function






What is the domain of a function?

  • The domain of a function is the values of x (the “input”) are allowed to take
  • For some functions, x cannot be certain values
    • eg. f(x) = 1 / xx cannot take the value 0


  • Other times we may choose to restrict the values of x
    • eg. The function g(x) = 5 – 2x2 is used to model the height of water throughout the day where x indicates timeIt may make sense to limit x so it only covers a 24 hour period


  • Inequalities are used to describe the values x can take
  • Any exclusions are usually indicated using the “not equal to” symbol ()




What is the range of a function?

  • The range of a function is the values of f (the “output”) that could occur
  • Some functions can never take certain values, regardless of the value of x
    • eg. f(x) = x2f, a squared (real) value, cannot be negative
    • eg2. g(x) = 1 / xg can never be zero (because numerator cannot be 0)


  • The range of a function can be influenced by its domain
  • As with the domain, inequalities are used to describe the values a function can take and “not equal to” () is used for any exclusions




How do I solve problems involving the domain and range



  • You need to be able to identify and explain any exclusions in the domain of a function
  • You need to be able to deduce the range of a function from its expression and domain
  • You may also be asked to sketch a graph of a function
    • This could involve sketching parts of familiar graphs that are restricted because of the domain and exclusions






Domain and range of inverse functions

  • Make sure you are familiar with inverse functions, denoted by f-1g-1, etc.
  • The range of a function is the domain of the inverse function
  • The domain of a function is the range of the inverse function



Exam Tip

A graph of the function can help “see” both the domain and range of function, and a sketch can help if you have not been given a diagram.

Worked Example