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In this video, we’ll learn how to use a function machine to calculate outputs, inputs, and missing operations and how function machines link to writing algebraic expressions and solving equations.
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A function machine is a way of writing rules using a flow diagram.
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These flow diagrams might only contain one operation such as multiply or add.
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But they can equally contain as many operations as is required.
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When inputting values to function machines, you go from left to right and apply the operations in order.
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Let’s see what that might look like.
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Use the function machine to find the output when the input is four.
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Here, we have a function machine represented by two operations.
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We follow the directions of the arrow, so left to right.
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And it tells us that we take an input, add two to it, then multiply by three, and we get the output.
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We want to work out the output when the input is four.
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So, let’s follow these steps.
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Since our input is four, we do four plus two, which is six.
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We then multiply this value by three to get the output.
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Six multiplied by three is 18.
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So, with this function machine, when the input is four, the output is 18.
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We’ll now look at how we can work out unknown operations, given a function machine.
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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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Here, we have the same function machine with two different inputs.
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When we input a value of seven, do something to it, then add seven, we get 21.
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And when we input a value of three, do something to it, and add seven, we get 13.
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So, we could use a bit of trial and error to find the missing input.
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But if we’re clever, we can save ourselves a little bit of time.
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Notice how the final operation is add seven.
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So, can we work out what the middle value would have been before this operation was applied?
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Well, yes, we’re going to perform an inverse or opposite operation.
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The opposite to adding seven is to subtract seven.
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So, we begin by subtracting seven from each of our outputs.
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21 minus seven is 14.
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So, the intermediate value in our first function machine must be 14.
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13 minus seven is six.
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So, the intermediate step when we use an input of three is six.
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We can now ignore the second operation.
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And we see we have a much simpler function machine with an input of seven and three and outputs of 14 and six, respectively.
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Can you spot what single operation has been applied to both the seven and the three to get their respective outputs?
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In fact, they’ve both been multiplied by two.
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Seven times two is 14, and three times two is six.
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And so, the missing operation here is multiplied by two.
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Let’s check this solution by running seven back again through the full machine.
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We multiply seven by two to get 14.
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We then add seven to the 14, and we get 21, which is the output we were expecting.
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So, the missing operation is times two.
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In our next example, we’ll look at the link between function machines and algebraic expressions.
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The diagram below shows the operation of a number machine.
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Which of the following expressions describes 𝑘, the output of the number machine?
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Is it (A) 𝑐 plus 20 minus 15 times two, (B) 𝑐 plus 20 times two minus 15, (C) 𝑐 plus 20 minus 15 times two?
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Is it (D) 𝑐 minus 15 times two plus 20 or (E) 𝑐 plus 20 times two minus 15?
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Here, we have a function machine representing three operations.
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We input a number and then follow the directions of the arrow, so left to right.
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And it tells us that we add 20, timesed by two, and then subtract 15 to get our output.
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Now, we want to evaluate the output when the input is 𝑐.
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So, let’s follow those steps.
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And we won’t panic that we have a letter instead of a number; we’ll treat 𝑐 just as if it is a number.
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We begin by taking 𝑐 and adding 20 to it.
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That gives us 𝑐 plus 20.
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We then multiply by two.
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Now, a common mistake here would be to write 𝑐 plus 20 times two as shown.
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But this would mean only the 20 is being multiplied.
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In fact, we want the entirety of 𝑐 plus 20 to be multiplied by two.
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So, we add a pair of parentheses.
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It’s 𝑐 plus 20 all multiplied by two.
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Our final step is to subtract 15.
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We don’t need to add any further parentheses since the order of operations tells us to multiply before subtracting.
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And so, we have our expression for 𝑘.
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It’s 𝑐 plus 20 all multiplied by two minus 15.
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And the correct answer that matches this is (E).
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Note that it might be more usual to write this as two times 𝑐 plus 20 minus 15.
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In this case though, either way will give us the correct answer.
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In our next example, we’ll consider how we can find an input, given a specific output.
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Use the function machine to find the input which gives an output of 21.
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Here, we have a function machine represented by two operations.
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We follow the directions of the arrow, so left to right.
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And what this tells us is we take an input, add two to it, then multiply by three, and we get our output.
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Now, we’re given a specific output.
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So, to work out the input, given this output, we need to go backwards.
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We need to reverse this.
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We’re going to take our output of 21 and do the opposite, the inverse operation, of multiplying by three.
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The opposite of multiplying by three is dividing by three.
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And 21 divided by three is seven.
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Next, we take our seven, and we do the inverse or the opposite of adding two.
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The opposite of adding two is subtracting two.
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So, we subtract two from this seven to give us five.
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And the input must, therefore, have been equal to five.
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Now, of course, we can check our answer by running this input through the original machine and checking we do indeed get 21.
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We add two to the input, which is five, so five add two, which is seven.
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And we then multiply that seven by three, which does indeed give us 21 as required.
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The input which gives an output of 21 is five.
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In fact, what we’ve just done is represented the inverse of our function machine.
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The inverse of our function machine is divide by three and then subtract two.
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So, let’s have a look at another example.
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Callie has drawn a function machine as shown in the figure.
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She wants to create an inverse function machine.
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Fill in the missing operations.
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Hence, solve the equation three 𝑥 minus four equals 62.
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So, we have a simple function machine represented by two operations.
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Now, we shouldn’t worry about the terminology here.
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We take the input, which we’re saying is 𝑥.
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We multiply it by three and then subtract four.
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We then get the output, which is some function in 𝑥, 𝑓 of 𝑥.
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She wants to create an inverse.
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Remember, that means opposite function machine.
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So, we need to work out the missing operations that will take us from the function 𝑓 of 𝑥 back to the original input.
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Remember, we’re going to be working backwards, so we’ll begin by looking at the second operation.
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The second operation is to subtract four.
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So, what is the opposite of subtracting four?
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The opposite of subtracting four is adding four.
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So, to reverse this step, we need to put a plus four in this box.
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Now, let’s consider the first operation.
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Remember, this will be the second operation in our inverse function machine since we’re working backwards.
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The first operation here is to multiply by three.
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So, what’s the opposite of multiplying by three.
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The opposite of timesing by three is dividing by three.
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So, the inverse function machine is as shown.
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We take our original function or our output.
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We add four to it then divide by three, and we get our original input.
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The second part of this question says, hence, solve the equation three 𝑥 minus four equals 62.
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Now, this might seem like a little bit of a leap.
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But let’s just go back to our original function machine.
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Let’s take our input of 𝑥 and times it by three.
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When we do, we get three times 𝑥, which is three 𝑥.
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The next operation is to subtract four.
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When we subtract four, we get our function 𝑓 of 𝑥.
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Well, three 𝑥 take away four is just three 𝑥 minus four, as shown.
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Notice, this is the same as the algebraic expression in our equation.
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And so, what this question’s really asking us is, what value of 𝑥 for the input will give us 62 as our output?
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And so, this means we can use our inverse function machine to calculate this value.
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Remember, 62 is our output.
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We’re going to follow the arrows on our inverse function machine.
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And this means we’re going to add four to 62.
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That gives us 66.
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We then follow the next arrow.
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And that tells us to take our value of 66 and divide it by three.
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Well, 66 divided by three is 22.
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And so, this tells us that 𝑥 must be 22 for an output of 62.
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So, we can check this result by putting 𝑥 back into our original function machine.
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This time, we let our input be 22.
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And then, we multiply that by three.
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That gives us 66.
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Next, we take away four from 66, and we get 62 as required.
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The solution to the equation three 𝑥 minus four equals 62 is, therefore, 𝑥 equals 22.
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In this video, we’ve seen that a function machine is a way of representing a series of rules or operations using a flow diagram.
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We’ve seen that when inputting values into function machines, you go from left to right and apply the operations in order.
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Very occasionally, function machines will be top down, so you’ll start at the top and apply the operations in a downwards manner.
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We also saw that we can reverse these steps to find inputs, given a specific output.
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This is called finding the inverse function machine.