Introduction to Functions

A function machine (often called a number machine) is a way of showing a function in a diagram
- A function machine starts with an input that leads into a series of boxes each labelled with a single mathematical function
  - The number of labelled boxes is the same as the number of individual mathematical operations in the function
- An arrow leading out of the final box gives the output of the function

A function machine showing an input leading to a function x3 leading to another function -2 leading to the output

You will need to be able to use function machines to
- Generate an output from a given input
  - Starting from the left hand side, apply the mathematical operation in the first box to the given input
  - Apply the mathematical operation in the next box to the result
  - Write down the output value after all operations have been completed
- Find an input from a given output
  - Redraw the function machine with the opposite mathematical operations in the boxes and the arrows pointing in the other direction
  - Starting from the right hand side, apply the opposite mathematical operation to the given output
  - Apply the opposite mathematical operation in the next box to the result
  - Write down the input value after all opposite operations have been completed
You will need to be able to draw input/output diagrams
Function questions may involve different mathematical topic areas such as sequences or coordinates or simple algebra

Here is a number machine

A function machine showing the operations +5 and x3

(a)

Work out the output when the input is 6.

The input is 6
Starting from the left hand side apply the first operation (+5) to the input

$6 \to 6 + 5 \to 11$

Use the output of the first operation as the input for the second operation

$\begin{array}{rcl} 11 & \to & 11 \times 3 \to 33 \end{array}$

33

(b)

Work out the input when the output is 21.

Sketch a new function machine with the opposite mathematical operations in the boxes and the arrows reversed
Work from the right hand side

An inverted function machine

Apply the opposite mathematical operation in the first box from the right hand side (÷3) to the output

$21 \to 21 \div 3 \to 7$

Apply the opposite mathematical operation in the next box (-5) to the result

7 \to 7 - 5 \to 2

2

Introduction to Functions (Edexcel GCSE Maths: Foundation)