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