Relations and Functions

About this unit

This unit builds the language that the rest of higher mathematics uses. A relation links elements of one set with elements of another, and a function is a special relation in which every input has exactly one output. You study how relations behave (reflexive, symmetric, transitive), how functions behave (one-one, onto, invertible), and how the inverse trigonometric functions are defined by restricting the domains of sine, cosine, tangent and the others. These ideas return in calculus, where derivatives and integrals of inverse trigonometric functions are used constantly.

Key ideas

• A relation on a set A is any subset of A × A.
• Equivalence relation: reflexive, symmetric and transitive together; it splits the set into equivalence classes.
• One-one (injective) and onto (surjective) functions; a bijection is both.
• Composition gof means "apply f first, then g".
• A function has an inverse exactly when it is a bijection.
• Inverse trigonometric functions use principal value branches, such as [−π/2, π/2] for sin⁻¹.

How to prepare

Memorise the domain and principal range of all six inverse trigonometric functions. Practise testing small relations for each property by listing ordered pairs. Expect questions that ask you to classify a relation or function, compute a composition, or find a principal value.