FA7, HA7 - Inputs and outputs

FA7, HA7 - Inputs and outputs

Turn an expression into a rule that acts on an input and produces an output. FA7 and this part of HA7 connect algebra to number machines: read the operations, calculate outputs in the correct order, and work backwards to recover a missing input.

Functions as rules

Two machines use the operations “multiply by 33” and “add 44”. One multiplies first; the other adds first. Will an input of 55 produce the same output?

An input is the starting number. An output is the result after the rule has acted on it. The expression 3x+43x+4 describes the output when the input is xx: multiply the input by 33, then add 44.

Function

A function is a rule that gives exactly one output for each allowed input. A number machine displays the operations in the order they act.

The input letter is a placeholder for a number. In 3x+43x+4, the multiplication is done before the addition. In 3(x+4)3(x+4), the brackets make the addition happen first. The same operations in a different order can give different functions.

Worked example 1: reading an expression as a machine

Question. Describe the machine for the output 3x+43x+4. Find its output when the input is 55.

Plan. Read the expression using the usual order of operations.

The machine first multiplies the input by 33, then adds 44.

5×3=15multiply first15+4=19then add four\begin{aligned} 5\times3&=15 &&\text{multiply first}\\ 15+4&=19 &&\text{then add four} \end{aligned}

Answer. Multiply by 33, then add 44; the output is 1919.

Check. Substitute directly into the expression: 3(5)+4=15+4=193(5)+4=15+4=19. Reversing the steps also returns 55: 19−4=1519-4=15, 15÷3=515\div3=5.

Keep both operations, but put the addition inside brackets.

Worked example 2: brackets change the order

Question. Describe the machine for the output 3(x+4)3(x+4). Find its output when the input is 55.

Plan. Complete the bracket before multiplying.

5+4=9addition is inside the bracket9×3=27multiply the whole bracket\begin{aligned} 5+4&=9 &&\text{addition is inside the bracket}\\ 9\times3&=27 &&\text{multiply the whole bracket} \end{aligned}

Answer. Add 44, then multiply by 33; the output is 2727.

Check. Expanding gives 3(x+4)=3x+123(x+4)=3x+12. At x=5x=5, 15+12=2715+12=27. This differs from 1919, so the opening machines do not have the same output.

You can translate other expressions in the same way. For x/3+4x/3+4, divide by 33, then add 44. For (x+4)/3(x+4)/3, add 44, then divide the whole result by 33.

Watch out

“Multiply by 33, then add 44” gives 3x+43x+4, not 3(x+4)3(x+4). Brackets change which quantity is multiplied.

Read the operations in a new expression.

Evaluating outputs

To calculate an output, pass the input through every operation once, in order. The result of one step becomes the input to the next.

You already know substitution: replace the letter by the whole signed input. Writing the intermediate value makes it easier to see where an error occurs. This is especially useful with negative or fractional inputs.

Worked example 3: following a two-step rule

Question. A machine multiplies by 44, then subtracts 77. Work out the output when the input is 66.

Plan. Multiply the input first and subtract from the intermediate result.

6×4=24first operation24−7=17second operation\begin{aligned} 6\times4&=24 &&\text{first operation}\\ 24-7&=17 &&\text{second operation} \end{aligned}

Answer. The output is 1717.

Check. Undoing the steps gives 17+7=2417+7=24, then 24÷4=624\div4=6, recovering the given input.

Change only the sign of the input. The operations themselves stay the same.

Worked example 4: using a negative input

Question. The same machine multiplies by 44, then subtracts 77. Work out its output when the input is −6-6.

Plan. Keep the negative sign during the multiplication, then subtract 77.

(−6)×4=−24negative times positive is negative−24−7=−31move seven further below zero\begin{aligned} (-6)\times4&=-24 &&\text{negative times positive is negative}\\ -24-7&=-31 &&\text{move seven further below zero} \end{aligned}

Answer. The output is −31-31.

Check. −31+7=−24-31+7=-24 and −24÷4=−6-24\div4=-6. The recovered input has the required negative sign.

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