Assumed knowledge

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

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 $$g(x) = \dfrac{x}{2}$$.
We shall develop a sufficiently general concept of inverses to cover the example of $$x^2$$ and $$\sqrt{x}$$, where we have $$(\sqrt{x})^2 = x$$ and $$\sqrt{x^2} = x$$, but with some restrictions on $$x$$.

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.

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