Functions, relations and graphs

A relation is any set of ordered pairs (x, y). It might be given as a list, a table, a graph or a rule such as x² + y² = 25. A function is a special relation in which every x-value in the domain is paired with exactly one y-value. Put another way, the input never produces two different outputs.

Graphically we test this with the vertical line test: if any vertical line meets the graph more than once, the relation is not a function. The circle x² + y² = 25 fails, because the line x = 3 meets it at (3, 4) and (3, −4). The parabola y = x² passes, but the sideways parabola x = y² fails, because x = 4 gives y = 2 and y = −2.

Functions are described using function notation. Writing f: D → R, f(x) = rule, records three things at once: the name of the function (f), the domain D (the set of allowed inputs), the co-domain R (the set the outputs are drawn from) and the rule. The range is the set of outputs actually produced, so it is always a subset of the co-domain. For example, f: R → R, f(x) = x² has co-domain R but range [0, ∞).

The expression f(3) means the output when the input is 3, and solving f(x) = 7 means finding the inputs that give an output of 7. These are different tasks, and confusing them is a common error.

When a rule is given without a domain, we assume the maximal (natural or implied) domain: the largest set of real numbers for which the rule produces a real output. Two restrictions appear constantly in Year 11. You cannot divide by zero, so any value making a denominator zero is excluded. You cannot take the square root (or any even root) of a negative number, so the expression under the root must be at least zero.

Domains and ranges are written in interval notation: square brackets include an endpoint, round brackets exclude it, and ∞ always takes a round bracket. The notation R \ {3} means all real numbers except 3.

A function can be represented in several ways. A rule is compact and exact; a table shows selected values and is useful for plotting; a graph shows behaviour at a glance. Moving between them is a core skill: from a rule you build a table, from a table you sketch a graph, and from a graph you read off approximate values, the domain and the range.

When reading a range from a graph, look at the lowest and highest y-values the curve reaches and check whether the endpoints are included (closed dot) or excluded (open dot).