OT1: Mathematical argument, language and proof: Understand and use the definition of a function; domain and range of functions.

The idea

A function is a rule that assigns to each element of a set exactly one output. The domain is the set of allowed inputs, and the range is the set of actual outputs. For example, f(x) = 1/(x − 2) has domain x ≠ 2 because division by zero is undefined, and its range is all real y except 0. For g(x) = √(x − 3), the domain is x ≥ 3 and the range is y ≥ 0. You must state domains explicitly when they are restricted, use correct notation such as f: x ↦ x² for x ∈ ℝ, and distinguish the range from the codomain. When combining or inverting functions, check that the domain makes each step valid, and remember that a function must pass the vertical line test when shown as a graph.

What earns the marks

• States a correct domain by identifying values that make a function undefined, such as division by zero or a negative square root.
• Determines the range from the rule and domain, using algebraic reasoning or a sketch of the graph.
• Uses function notation correctly, including f(x), f: x ↦ ... and substitution of particular values.
• Distinguishes between the domain, the codomain and the range when these are not all the same set.
• Applies domain restrictions when forming composite functions or inverses, checking that inputs lie within the required domain.
• Interprets a graph to identify domain and range, including open and closed endpoints.

Common mistakes and how to put them right

• Assuming the domain is always all real numbers.

  Put it right: For f(x) = 1/x the domain excludes x = 0, and this must be stated.

• Confusing range with codomain.

  Put it right: The codomain is the set the outputs are declared to lie in, while the range is the set of outputs actually attained.

• Giving the range of a quadratic as all real numbers.

  Put it right: For f(x) = x² with domain ℝ the range is y ≥ 0, not ℝ.

• Forgetting to restrict the domain when finding an inverse, so the inverse is not a function.

Exam tips

• Write the domain as a condition on x, such as x ≠ 2 or x ≥ 3, rather than only describing it in words.
• Sketch the graph to find the range; the lowest or highest point and any asymptotes usually determine the endpoints.
• Check each restriction separately when a function combines a fraction, a root and a logarithm, then combine the conditions.
• When asked for the range, give it as a set or inequality in terms of y, and justify it briefly from the rule or graph.