CHAPTER 02

Functions and Elementary Models

Understand how mathematics describes relationships between quantities.

\[ x \longrightarrow f \longrightarrow f(x) \]

FUNCTIONS

Functions as mathematical relationships

What Is a Function?

Understand a function as a rule or mapping that assigns outputs to inputs.

Function Notation

Understand \(f(x)\), function names, inputs, outputs and evaluation.

Domain, Codomain and Range

Understand which inputs are allowed and which outputs a function can produce.

Linear Functions

Constant rates of change, straight lines, gradients and intercepts.

Quadratic Functions

Parabolas, roots, turning points and the structure of quadratic relationships.

Polynomial Functions

Understand polynomial expressions, degree, roots and their graphs.

Rational Functions

Functions formed from ratios of polynomials and the restrictions they introduce.

Exponential Functions

Multiplicative change, exponential growth and exponential decay.

Logarithmic Functions

Logarithms as inverse operations of exponentiation and their mathematical meaning.

Trigonometric Functions

Sine, cosine and tangent as functions describing angles and periodic behaviour.

Composite Functions

Understand how the output of one function can become the input of another.

Inverse Functions

Understand how an inverse function reverses the action of a function.

Transformations of Functions

Translations, reflections and scaling of functions and their graphs.

Where this is leading

\[ x \rightarrow f(x) \rightarrow f(x+\Delta x) \rightarrow \frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Once functions are understood, we can ask how their outputs change when their inputs change. This leads naturally to rates of change, limits and derivatives.