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

    • {1, 2, 3, ...} maps to {1, 8, 27, ...}

  • 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

    • {±1, ±2, ±3, ...} map to {1, 4, 16, ...}

  • 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

    • {1, 4, 16, ...} maps to {±1, ±2, ±3, ...}

  • 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

    • e.g. the factors of {2, 3} are {1, 2, 3}

    • one input, e.g. {2}, has many outputs, e.g. {1, 2}

    • one output, e.g. {1}, has many inputs, e.g. {2, 3}

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

• This means a function can be

  • one-to-one

  • or many-to-one

• A sketch of the function must pass the vertical line test

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

    • e.g. and any vertical line pass the test

What notation is used for functions?

• Functions are denoted using letters (such as  etc)

  • If is the input

    • then is the output of the function

• e.g. if when, then 

What are the domain and range of a function?

• The domain of a function is the set of all 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

  • Domains are expressed in terms of

    • e.g.

• The range of a function is the set of all outputs

  • The range depends on the domain

  • Ranges are expressed in terms of

    • e.g.

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

  • corresponds to the coordinates (2, 5)

What sets of numbers do I need to know?

• 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  and  are integers and

  • 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. 

    • so

    • whereas

• The region for the individual functions cannot overlap

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

  • In the example above the function is

    • continuous at  as both sides match: 

    • not continuous at  as