Language of Functions (DP IB Analysis & Approaches (AA)): Revision Note

Language of Functions

What is a mapping?

• A mapping transforms one set of values (inputs) into another set of values (outputs)

• Mappings can be:

  • One-to-one

    • Each input gets mapped to exactly one unique output

    • No two inputs are mapped to the same output

    • For example: A mapping that cubes the input

  • Many-to-one

    • Each input gets mapped to exactly one output

    • Multiple inputs can be mapped to the same output

    • For example: A mapping that squares the input

  • One-to-many

    • An input can be mapped to more than one output

    • No two inputs are mapped to the same output

    • For example: A mapping that gives the numbers which when squared equal the input

  • Many-to-many

    • An input can be mapped to more than one output

    • Multiple inputs can be mapped to the same output

    • For example: A mapping that gives the factors of the input

What is a function?

• A function is a mapping between two sets of numbers where each input gets mapped to exactly one output

  • The output does not need to be unique

• One-to-one and many-to-one mappings are functions

• A mapping is a function if its graph passes the vertical line test

  • Any vertical line will intersect with the graph at most once

What notation is used for functions?

• Functions are denoted using letters (such as  etc)

  • A function is followed by a variable in a bracket

  • This shows the input for the function

  • The letter is used most commonly for functions and will be used for the remainder of this revision note

• represents an expression for the value of the function when evaluated for the variable

• Function notation gets rid of the need for words which makes it universal

  • when can simply be written as 

What are the domain and range of a function?

• The domain of a function is the set of values that are used as inputs

• A domain should be stated with a function

  • If a domain is not stated then it is assumed the domain is all the real values which would work as inputs for the function

  • Domains are expressed in terms of the input

    • 

• The range of a function is the set of values that are given as outputs

  • The range depends on the domain

  • Ranges are expressed in terms of the output

    • 

• To graph a function we use the inputs as the x-coordinates and the outputs as the y-coordinates

  • corresponds to the coordinates (2, 5)

• Graphing the function can help you visualise the range

• Common sets of numbers have special symbols:

  • represents all the real numbers that can be placed on a number line

    • means is a real number

  • represents all the rational numbers where a and b are integers and b ≠ 0

  • represents all the integers (positive, negative and zero)

    • represents positive integers

  • represents the natural numbers (0,1,2,3...)

What are piecewise functions?

• Piecewise functions are defined by different functions depending on which interval the input is in

  • E.g. 

• The region for the individual functions cannot overlap

• To evaluate a piecewise function for a particular value

  • Find which interval includes 

  • Substitute into the corresponding function

• The function may or may not be continuous at the ends of the intervals

  • In the example above the function is

    • continuous at  as 

    • not continuous at  as