Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.312NA
uses function notation to describe and sketch functions
Content
 Students:
 Describe, interpret and sketch functions
 define a function as a rule or relationship where for each input value there is only one output value, or that associates every member of one set with exactly one member of a second set
 use the vertical line test on a graph to decide whether it represents a function
 explain whether a given graph represents a function (Communicating, Reasoning)
 decide whether straightline graphs always, sometimes or never represent a function (Reasoning)
 use the notation \(f(x)\)
 use \(f(c)\) notation to determine the value of \(f(x)\) when \(x=c\)
 find the permissible \(x\) and \(y\)values for a variety of functions (including functions represented by straight lines, parabolas, exponentials and hyperbolas)
 determine the inverse functions for a variety of functions and recognise their graphs as reflections of the graphs of the functions in the line \(y=x\)
 describe conditions for a function to have an inverse function (Communicating, Reasoning)
 recognise and describe the restrictions that need to be placed on quadratic functions so that they have an inverse function (Communicating, Reasoning)
 sketch the graphs of \(y=f(x)+c\) and \(y=f(xa)\), given the graph of \(y=f(x)\)
 sketch graphs to model relationships that occur in practical situations and explain the relationship between the variables represented in the graph (Communicating)
 consider a graph that represents a practical situation and explain the relationship between the two variables (Communicating, Reasoning)
Purpose/Relevance of Substrand
Functions are very important concepts in the study of mathematics and its applications to the real world. They are used extensively where situations need to be modelled, such as across science and engineering, and in business and economics. There are many ways to represent a function. A formula may be given for computing the output for a given input. Other functions may be given by a graph that represents the set of all paired inputs and outputs. In science, many functions are given by a table that gives the outputs for selected inputs. Functions, such as inverse functions, can be described through their relationship with other functions.