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Functions (DP IB Maths: AI HL)
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
- 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...)
- represents all the real numbers that can be placed on a number line
Examiner 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
Worked example
For the function :
a)
write down the value of .
b)
find the range of .
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Piecewise Functions
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
Worked example
For the piecewise function
,
a)
find the values of .
b)
state the domain.
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