Assumed knowledge

• Familiarity with elementary set theory, as discussed in the TIMES module Sets and Venn diagrams (Years 7–8).
• Familiarity with the algebraic techniques described in the TIMES module Formulas (Years 9–10).
• The content of the modules Algebra review and Coordinate geometry.
• Basic knowledge of the modules Trigonometric functions and circular measure and Exponential and logarithmic functions.

Motivation

The expression \(y=x^2\), which links the two pronumerals \(x\) and \(y\), can be thought of in several different ways.

In the early years of school, we seek pairs of values, such as \((x,y) =(3,9)\), which satisfy the equation. These ordered pairs determine a graph. In this case, the graph is the standard parabola.

A formula is an equation relating different quantities using algebra. So \(y=x^2\) is also a formula. In fact, \(y=x^2\) is an example of a function, in the sense that each value of \(x\) uniquely determines a value of \(y\).

In this module, we will study the concept of a function. The formula \(A=\pi r^2\) gives \(A\) as a function of \(r\). The formula \(V=\pi r^2h\) expresses \(V\) as a function of the two variables \(r\) and \(h\). In this module, we will only consider functions of one variable, such as the polynomial

\[ y = (x-1)(x-2)(x-3)(x-4). \]

A clear understanding of the concept of a function and a familiarity with function notation are important for the study of calculus. The use of functions and function notation in calculus can be seen in the module Introduction to differential calculus.

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