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
- One-to-one
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...)
Exam Tip
- Questions may refer to "the largest possible domain"
- This would usually be
unless natural numbers, integers or quotients has already been stated
- There are usually some exceptions
- e.g.
for functions involving a square root (so the function can be 1-to-1 and have an inverse)
- e.g.
for a reciprocal function with denominator x-2
- e.g.
- This would usually be
Worked Example
For the function :
a)
write down the value of
.
b)
find the range of
.
Piecewise Functions
What are piecewise functions?
- Piecewise functions are defined by different functions depending on which interval the input is in
- E.g.
- 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
- Find which interval includes
Worked Example
For the piecewise function
,
a)
find the values of
.
b)
state the domain.