A function machine takes an input, carries out some operations and then gives us an output. An example of a function machine is shown below:

**Input of 2**

Let’s now find what the output is if the input is 2.

We find out what the output is when the input is 2 by placing 2 into the function machine. The first operation in the function machine is “multiply by 3”, so we multiply 2 (our input) by 3.

This gives us 6.

We now complete the second operation in our function machine, which is “add 5”; we add 5 to 6.

We now complete the second operation in our function machine, which is “add 5”; we add 5 to 6.

We now have our output (11). Therefore, when the input is 2, the output is 11.

Let’s now find what the output is for our function machine when the input is 4.

We find the output when the input is 4 by placing 4 into the function machine. The first operation in the function machine is “multiply by 3”, so we multiply 4 (our input) by 3.

**Input of 4**Let’s now find what the output is for our function machine when the input is 4.

We find the output when the input is 4 by placing 4 into the function machine. The first operation in the function machine is “multiply by 3”, so we multiply 4 (our input) by 3.

This gives us 12.

We now complete the second operation in our function machine, which is “add 5”; we add 5 to 12.

We now complete the second operation in our function machine, which is “add 5”; we add 5 to 12.

We now have our output (17). Therefore, when the input is 3, the output is 17.

**Given the output**

It may be the case that we are given the output for a function machine and are asked to find what the input was. We answer questions like this by carrying out the opposite operations from right to left.

For example, for the function below, what is the input when the output is 23.

We find the input from the output by carrying out the opposite operations working from right to left. The first operation that we come across in the function machine when working from right to left is “add 5”. The opposite operation for adding is taking. Therefore, this operation when working from the output to input becomes “take 5”; we take 5 from our output (23).

This gives us 18.

We now complete the opposite of the next operation to the left, which is “multiply by 3”. The opposite of multiplying is dividing. Therefore, this operation when working from the output to the input becomes “divide by 3”; we divide 18 by 3.

We now complete the opposite of the next operation to the left, which is “multiply by 3”. The opposite of multiplying is dividing. Therefore, this operation when working from the output to the input becomes “divide by 3”; we divide 18 by 3.

We now have our input (6). Therefore, when the output is 23, the input is 6.

The output for the function below is 56. Find the input.

**Output of 56**The output for the function below is 56. Find the input.

We find the input from the output by carrying out the opposite operations working from right to left. The first operation that we come across in the function machine when working from right to left is “add 5”, and the opposite of this operation is to “take 5”; we take 5 from the output (56).

This gives us 51.

We now carry out the opposite for the next operation that we come across to the left. The next operation that we come across is “multiply by 3”, and the opposite of this operation is “divide by 3”; we divide 51 by 3.

We now carry out the opposite for the next operation that we come across to the left. The next operation that we come across is “multiply by 3”, and the opposite of this operation is “divide by 3”; we divide 51 by 3.

We now have our input (17). Therefore, when the output is 56, the input is 17.

**Example 2**

We have the function machine below.

Answer the following:

a) Find the output when the input is 9

b) Find the output when the input is -2

c) Find the input when the output is 60

The first part of the question asks us to find the output for the function machine when the input is 9. We are able to do this by placing 9 into the function machine. The first operation in the function machine is “take 6”, so we take 6 from our input (9).

a) Find the output when the input is 9

b) Find the output when the input is -2

c) Find the input when the output is 60

**Part a**The first part of the question asks us to find the output for the function machine when the input is 9. We are able to do this by placing 9 into the function machine. The first operation in the function machine is “take 6”, so we take 6 from our input (9).

This gives us 3.

We now complete the second operation in our function machine, which is “multiply by 4”; we multiply 3 by 4.

We now complete the second operation in our function machine, which is “multiply by 4”; we multiply 3 by 4.

We now have our output (12). Therefore, when the input is 9, the output is 12.

The second part of the question asks us to find the output for the function machine when the input is -2. We are able to do this by placing -2 into the function machine. The input (-2) is a negative, so we need to be very careful when we are carrying out the operations. The first operation in the function machine is “take 6”, so we take 6 from our input (-2).

**Part b**The second part of the question asks us to find the output for the function machine when the input is -2. We are able to do this by placing -2 into the function machine. The input (-2) is a negative, so we need to be very careful when we are carrying out the operations. The first operation in the function machine is “take 6”, so we take 6 from our input (-2).

This gives us -8.

We now complete the second operation in our function machine, which is “multiply by 4”; we multiply -8 by 4.

We now complete the second operation in our function machine, which is “multiply by 4”; we multiply -8 by 4.

We now have our output (-32). Therefore, when the input is -2, the output is -32.

The final part of the question asks us to find the input of the function when the output is 60. We find the input from the output by carrying out the opposite operations working from right to left. The first operation that we come across in the function machine when working from right to left is “multiply by 4”, and the opposite of this operation is “divide by 4”; we divide the output (60) by 4.

**Part c**The final part of the question asks us to find the input of the function when the output is 60. We find the input from the output by carrying out the opposite operations working from right to left. The first operation that we come across in the function machine when working from right to left is “multiply by 4”, and the opposite of this operation is “divide by 4”; we divide the output (60) by 4.

This gives us 15.

We now carry out the opposite for the next operation that we come across to the left. The next operation that we come across is “take 6”, and the opposite of this operation is “add 6”; we add 6 to 15.

We now carry out the opposite for the next operation that we come across to the left. The next operation that we come across is “take 6”, and the opposite of this operation is “add 6”; we add 6 to 15.

We now have our input (21). Therefore, when the output is 60, the input is 21.