Chapter 15: Functions, algebraic fractions and proof

The chapter problem: a two-step machine

A function machine takes an input, applies a rule, and gives an output. A machine first doubles its input and adds $3$. A second machine subtracts $3$ and halves its input. Predict what happens when you use both machines, then decide whether their order matters. You can test your prediction in Section 15.1 and explain it with algebra by the end of this chapter.

First contact: make a prediction using the input $4$. You can estimate or write a number; there is no mark for your first answer.

By the end of this chapter you will be able to

• find inverse and composite functions and state restrictions on their inputs;
• simplify algebraic fractions by factorising and cancelling common factors;
• add, subtract, multiply and divide algebraic fractions;
• distinguish an identity from an equation and use equivalent expressions;
• build algebraic proofs from definitions and valid steps.

Roadmap

1. 15.1 Functions: inverse and composite
2. 15.2 Simplifying algebraic fractions
3. 15.3 Operations with algebraic fractions
4. 15.4 Identities and equivalent expressions
5. 15.5 Algebraic proof
6. Review

Your turn to connect

Where have you seen a rule applied in stages, such as changing a price or converting a measurement? Write one example and explain how reversing the stages might help you check the result.

How to use this chapter

Each section has Learn, Practise with help, and Practise pages. The Review page returns to the two machines and mixes their ideas with earlier algebra. You can revisit a worked example whenever you need it.