Pure 2. Functions

A Level Mathematics 9709 Examinations 2023 – 2025 Pure 1

Why do we study functions?

Situations in the real world that can be modelled as functions:

• the temperature of a hot drink as it cools over time
• the height of a valve on a bicycle tyre as the bicycle travels along a horizontal road
• the depth of water in a conical container as it is filled from a tap
• the number of bacteria present after the start of an experiment
• Modelling these situations using appropriate functions enables us to make predictions about real-life situations, such as: How long will it take for the number of bacteria to exceed 5 billion?

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*Before we begin, a quick review of types of numbers

Q

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R

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Z

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N

1. Mapping fundamentals

A mapping is a rule which links items in one set (the inputs or objects) to items in another set (the outputs or images).

One-to-many mapping

Many-to-many mapping

1. Mapping fundamentals

In a many-to-one mapping,

• each input maps to one output
• some points in range are image of >1 in the domain

Inputs (or objects)

Outputs (or images)

In a one-to-one mapping,

• each input has exactly one output
• each output has exactly one input

Domain

Range

A function is a mapping where each point in the domain maps to one image point

• many-to-one
• one-to-one

2. Composite functions

= combination of 2 or more functions

3. Inverse functions

= undoes what the function does

*not every function has an inverse

4. Graph of a function and its inverse

- domain and range of f(x)?

- domain and range of inverse function?

- how do the 2 graphs look?

5. Transformations

a) Translation

b) Reflection

c) Stretches

The graph of y = af(x) is a stretch of the graph y = f(x) with stretch factor a parallel to the y-axis.

c) Stretches

Vertical transformations follow the ‘normal’ order of operations, as used in arithmetic.

Horizontal transformations follow the opposite order to the ‘normal’ order of operations, as used in arithmetic.

y = 2f(x) - 3 is a combination of two vertical transformations of y = f(x), hence the transformations follow the ‘normal’ order of operations.