Functions

Introduction

The Functions Chapter of the A level, usually in the Core 3 module, isn't a very difficult chapter conceptually. However there are many terms etc. that must be learnt in order to do well in the exam. Subtleties, such as in which order you must perform transformations, must be taken with care.

The Language of Functions

KEY TERM:

  • Mapping: An operation that associates each element of a given set (the domain) with one or more elements of a second set (the range).


But what does this really mean? Well in short, a mapping is a rule which shows you how to get from an input to an output. This can cover a whole range of things.

e.g.
MAPPING: Take the input, and add 4.




INPUT: 1
OUTPUT: 5

or

INPUT: 7
OUTPUT: 11



Mappings can be shown by using a diagram, here is one for the example mapping mentioned above:

Figure 1-An Example of a mapping



KEY TERMS:

  • Domain: All the possible input values for your mapping (for a function, your x values).
  • Range: All the possible output values for your mapping (for a function, your y values).

All the functions you've come across before in maths are examples of mappings.
You have an input (your x values), and you have a rule which tells you what to do in order to get to your output (your y values)!
But does that mean all mapping are functions?

NO!

There are four types of mappings. Two of which ARE functions, but the other two certainly aren't!


Types of Mapping

As fore-mentioned mappings come in four different flavors.

  • One-to-one
  • This sort of mapping has only one possible output for any given input. And each input has a unique output. For example the mapping in figure 1 is an example of a one-to-one mapping.

  • One-to-many
  • Many-to-one
  • Many-to-many

Transformations

Transformations- General case:

    Lets take f(x) as our original function and discover how changes to the function result in graphic transformations.


  • Stretches:
    Stretches parallel to both the x and y axes can be represented by transformations:


    This represents a stretch parallel to the y axis, scale factor C.


    Figure 2: Example of a stretch in the y direction



    This represents a stretch parallel to the x axis, scale factor 1/C.


    Figure 3: Example of a stretch in the x direction

Composite Functions

blah blah blah

Inverse Functions

blah blah blah

Inverse Trigonometric Functions

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Even, odd and Periodic Functions

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The Modulus Function

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