Jennifer has drawn a function machine with two different inputs as seen in the figure. Work out the missing operation that she has left out of the first box.
In this question, we have a function machine. A function machine is a diagram representing a machine which takes an input, applies a rule or an operation, and delivers the answer as an output. In this question, we’re looking at the same machine, which has given two different inputs, a seven and a three. There are two different rules, one of which we need to find out. And the input of seven gives an output of 21. And an input of three gives an output of 13. Let’s have a recap of how we use a function machine.
Let’s imagine we have a function machine where the rule is times five, or multiply by five. Given an input of three, when we apply the rule, that would be three times five, giving us an output of 15. However, if the same function machine gives us an output of 30 for an unknown number, we can use the rule to help us work this out. To find out what times five equals 30, we would apply the inverse operation, which in this case would be dividing by five. And since 30 divided by five gives us six, our input must have been six.
So, let’s now have a look at the function machine in the question with the first input of seven. We can see that the first rule was applied then another rule and the output was 21. We need to work out what the answer was before the rule plus seven was applied. We could think of this as the input into the second rule. Let’s apply the inverse operation of plus seven, which would be subtracting seven. So, since 21 take away seven is 14, we know that the number at this point must have been 14. And now we need to think, what must we do to seven to get to 14?
If we think of this in terms of a single operation, then there are two possibilities. The first one could be times two because seven times two would give us 14. And the second option could be plus seven because seven plus seven would give us 14. What we need to do now is try each of these operations with our second input to see which would give us the correct output of 13.
So, let’s start by checking our rule of plus seven. The input of three plus seven would give us a value of 10 since three add seven is 10. Next, we apply the rule of plus seven to our value 10, which would give us an answer of 17. So, the output of our function machine would be 17. However, we were looking for an output of 13. So, that means our rule plus seven must be incorrect.
So, let’s try our rule of times two instead. If we take our input of three and multiply by two, this will give us the value of six. Then, we apply our rule of plus seven to our value of six, which will give us an answer, or an output, of 13. And this does match the output we were asked for. So, now, we know that multiplying by two must be the missing operation in our function machine.