## Assumed knowledge

This module builds on the module Functions I. It assumes knowledge of that module and also the modules required for it:

- Algebra review
- Coordinate geometry
- Trigonometric functions and circular measure
- Exponential and logarithmic functions.

## Motivation

The importance of the concept of a function to modern mathematics was explained in the module Functions I. In this module, we shall develop five aspects of the theory.

- 1Arithmetic of functions.
- We shall define how to add, subtract, multiply and divide functions.
- 2Odd and even functions.
- In particular, we shall see that the order of composition is important, so that \(\sin(x^2)\) and \((\sin x)^2\) are completely different functions.
- 3Composition of functions.
- We shall see that many functions can be understood in terms of a geometrical transformation, such as a translation or a reflection.
- 4Geometrical transformations.
- We shall investigate which functions have inverses. For example, the functions in the following pairs are inverses of each other:
- 5Functions and their inverses.

- \(f(x) = x+2\) and \(g(x) = x-2\)
- \(f(x) = 2x\) and \(g(x) = \dfrac{x}{2}\).

All of these ideas are important in differential calculus and in curve sketching.

In Functions I, we covered:

- the concept of a function
- the difference between functions and relations
- the vertical-line test
- domains and ranges
- interval notation
- standard function notation.

In this module, we shall build on these ideas.