Function Notation

Expressions as input–output rules

An expression such as $2x+3$ can be read as a set of instructions. Start with an input $x$, multiply it by 2, then add 3. The result is the output $y$, so the relationship can be written $y=2x+3$. If the input is 5, the output is $2\times5+3=13$. The same rule works for other inputs: putting in $-1$ gives $2(-1)+3=1$.

A function machine makes the order visible. Each box changes the number it receives, and its result passes to the next box. Here is the rule $y=2x+3$ alongside the process that reverses it.

Working backwards: an inverse function

Suppose you know the output 13 and want to recover the input. The last thing the original rule did was add 3, so undo that first: $13-3=10$. Then undo the multiplication by 2: $10\div2=5$. The inverse process is therefore ‘subtract 3, then divide by 2’. For this rule, every output leads back to one input, so the inverse can also be described by the expression $(y-3)\div2$.

Reversing a multi-step rule means reversing both the operations and their order. For example, dividing 13 by 2 before subtracting 3 would not recover 5. Some rules cannot be reversed to a single input without a restriction: squaring gives the same output, 9, from both 3 and $-3$.

Joining rules: a composite function

A composite function is made by using the output of one function as the input to another. Imagine a first rule that doubles the input and adds 3, followed by a second rule that subtracts 4. Starting with 5, the first rule gives 13; the second then gives $13-4=9$. Taken together, the two rules give the output $2x+3-4=2x-1$.

The order matters. Subtracting 4 before doubling and adding 3 would give a different rule and, for an input of 5, a different output. You need not use special function notation here: a labelled input–output diagram or a clear description of the successive rules shows the process.

Input and output

Read $y=2x+3$ as input $x\;\rightarrow\;\times2\;\rightarrow\;+3\;\rightarrow\;y$. Follow the operations in order to find an output.

Inverse

Work backwards from the output, undoing operations in reverse order. The inverse of ‘$\times2$, then $+3$’ is ‘$-3$, then $\div2$’.

Composite

Apply the first function, then feed its output into the second. Changing their order can change the result. OCR does not require function notation for this topic.