## What you need to know

A function machine takes an input, applies a series of mathematical operations to it, and then outputs a result.

Example: Below is a function machine.

a) Work out the output we get when we input 7.

b) Work out the input needed to give 11 as an output.

c) Work out the output we get when we input $2x$.

a) As we can see, the function machine applies two operations: “$\times 4$” and “$+1$”, specifically in that order. So, with 7 as an input, first times by 4:

$7\times 4=28$

Then, subtract 1 to get the answer to be

$28-1=27$

b) To get an input from an output, we have to work backwards, doing the opposite operations in the opposite order. In this context, we consider adding to be the opposite of subtracting and multiplying to be the opposite of dividing.

Working backwards, we can see that the last operation is to subtract 1, so we must add 1 to our given output:

$11+1=12$

Then, the next operation is to multiply by 4, so we should divide by 4 to determine what the desired input would be.

$12 \div 4 = 3$.

CHECK: you can check your answer by putting 3 through the function machine and seeing if the output matches 11.

c) This time, we have to put some algebra into the machine, but the principle is example the same. Our first operation is multiplying by 4, so we get

$4\times 2x=8x$

Then, subtracting 1 we get

$8x-1$

You can see that we get an expression as an answer, not a number. This is to be expected any time you have to feed algebra through a function machine.

Example: Inputting 5 into the function machine below would give an output of 4.

a) Fill in the missing number in the function machine.

b) Using the function, work out the output produced by an input of $-6$.

a) So, if we input 5 into the function machine then we must first add 3:

$5+3=8$

Now, the question tells us that the result of dividing 8 by some number gives us 4. Clearly, dividing 8 by 2 would give 4, so the completed function machine is

b) Inputting $-6$, we must first add 3.

$-6+3=-3$

Now we know the second step is to divide by 2, so we get

$-3\div 2=-1.5$

## Example Questions

a) Inputting 35, we first multiply by 3:

$35 \times 3 = 105$

$105+15=120$

b) We must work backwards and do the opposite operations. So, first subtracting 15 from the given output, we get

$48-15=33$

Then, dividing by 3 we get

$33\div 3=11$

Meaning that 11 is the input required to give an output of 48.

$9\times 3=27$, so one operation would be multiplying by 3.

$9+18=27$, so a second operation would be adding 18.

We have to see what we get if we input $x$ into the function machine. First, times by 12:

$12\times x=12x$

Then, subtract 25 to get

$12x-25$

This is the output of inputting $x$ but we know the output is equal to $2x$, so we are left with the equation

$12x-25=2x$

Now we can solve this equation for $x$. Subtracting $2x$ from both sides, we get

$10x-25=0$

Adding 25 to both sides, we get

$10x=25$

Finally, dividing both sides by 10 we find $x$ to be

$x=25\div 10=2.5$