Notation, vocabulary and manipulation: where appropriate, interpret simple expressions as functions with inputs and outputs

The idea

A function is a rule mapping each input to exactly one output. In an expression such as 3x + 2, the input is x and the output is the value after applying the rule. You can write this as f(x) = 3x + 2, so f(4) means substitute x = 4: f(4) = 3 × 4 + 2 = 14. The phrase 'where appropriate' means you should interpret simple expressions as functions when it clarifies the relationship, but you may also work with the expression directly. For example, a function machine with input 5 and rule 'multiply by 3, then add 2' gives output 17. You must be able to read a simple expression as a function and identify its input and output.

What earns the marks

• Recognise that a function assigns exactly one output to each input.
• Interpret simple expressions as functions, identifying the rule applied.
• Substitute a numerical input into a simple expression and calculate the output.
• Interpret a function machine or mapping diagram as an expression with inputs and outputs.
• Explain the meaning of f(4) as the output when the input is 4.

Common mistakes and how to put them right

• Misunderstanding: thinking f(4) means f multiplied by 4.

  Put it right: f(4) means substitute 4 for x in the expression for f(x).

• Misunderstanding: treating the input and output as interchangeable.

  Put it right: The input is the value put into the function; the output is the result after applying the rule.

• Misunderstanding: applying operations in the wrong order.

  Put it right: Follow the order shown in the expression or function machine, using conventional priority of operations.

Exam tips

• When asked to process a function, write the substitution step clearly before calculating.
• Check whether the question uses function notation or a function machine, and keep the input and output distinct.
• For multiple-choice questions, test each option by substituting a simple input such as 2 or 3 to see which rule matches.