Functions can be a difficult concept to understand in mathematics. Functions machines are a helpful mathematical tool to assist in visualising functions in order to calculate inputs and outputs, to solve equations and find inverse and compound functions.
In this lesson you will learn what a function machine is and how to use function machines in mathematics.
Function Machines
What Are Function Machines?
A function machine is just a way to visualise a mathematical function as a rule or set of rules with an input and an output.
Functions can be quite abstract so drawing or writing them as a function machine can be helpful in understanding them.
Imagine placing an input into a ‘machine’ which then performs mathematical operations such as addition or multiplication. On the other side of the machine the output is returned. This is a function machine.
How to Find the Output of a Function Machine
- Consider the correct order of operations to perform.
- Draw the function machine (if this is not already given).
- Pass the input value into the function machine and perform the operations step-by-step to gain the output
Examples
How to Find the Input of a Function Machine
Sometimes you may be given the rules of a function machine and the output and asked to find the input. In this case you have to start from the output and perform the opposite operations in reverse order.
The opposite operations are called the inverse operations.
- For every step of the function machine determine the inverse operation. Eg. the inverse operation of +8 is -8 and the inverse operation of ÷ 11 is × 11.
- Start with the output of the function and perform the inverse operations in reverse order. Eg. if the function machine is +8 and then ÷ 11, you should × 11 and then + 8.
One Step Example
For a one-step function machine there is only one inverse operation to perform. So start with the output and perform the opposite operation (ie. the inverse operation).
Question: The operation is “add 12” and the output is 30. Find the input.
Answer: The inverse operation of “add 12” is “subtract 12”. So starting at 30 we need to subtract 12.
30 – 12 = 18 so the input was 18.
Two Step Example
For a two step function machine you need to think about the order of operations. The inverse operations must be performed in the opposite order. Eg. “Plus 5 and then multiply by 7″ becomes “divide by 7 and then subtract 5“.
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