Introduction to Functions
What is a function?
- A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers
- It may be thought of as a mathematical “machine”
- For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"
- Putting 3 in to the function would give 2 × 3 + 1 = 7
- Putting -4 in would give 2 × (-4) + 1 = -7
- Putting in would give
- The number being put into the function is often called the input
- The number coming out of the function is often called the output
What is a function machine?
- 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 starts with an input that leads into a series of boxes each labelled with a single mathematical function
How will I be expected to use functions?
- 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
- Generate an output from a given input
- 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
Worked example
Here is a number machine
The input is 6
Starting from the left hand side apply the first operation (+5) to the input
Use the output of the first operation as the input for the second operation
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
Apply the opposite mathematical operation in the first box from the right hand side (÷3) to the output
Apply the opposite mathematical operation in the next box (-5) to the result