Functions and admissible inputs

A function includes its input set

For f(x) = (x − 2)² on all real inputs, the range is [0,∞). Two inputs equidistant from 2 give the same output, so it is not one-to-one. For the same rule restricted to x ≥ 3, the range is [1,∞): the smallest allowed input is already to the right of the vertex. A rule alone does not determine the same range for every declared domain.

A mapping diagram has exactly one arrow leaving each allowed input. Different inputs may share an output. A graph has the same property when each vertical line meets it at most once. A whole circle is a relation with two y-values for many x-values, rather than a single y-function of x.

A displayed expression has real restrictions

For f(x) = 1/(x − 2), the denominator excludes x = 2 and the output can never be zero. Conversely every non-zero real output y is obtained by x = 2 + 1/y, so the range is all non-zero reals. A range claim needs both exclusion and attainability, not only a few sample points.

For g(x) = √(5 − x) on its natural real domain, require x ≤ 5 and outputs are non-negative. Every non-negative y is obtained with x = 5 − y², so its range is [0,∞). If an exercise supplies a smaller domain, use that given set rather than silently replacing it with the natural one.

Endpoints belong to the declared set

For h(x) = x² on −1 < x ≤ 2, the output 0 is attained at x = 0 and maximum 4 at x = 2, so the range is [0,4]. Although −1 is excluded, its output 1 is still attained at x = 1. Removing an input does not necessarily remove its output from the range.

Check your understanding 1 An excluded input can share an output

Find the range of f(x) = (x − 1)² on 0 < x ≤ 3. Explain the endpoints.

Your answer or drawing on paper

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The minimum 0 is attained at x = 1. The maximum 4 is attained at x = 3, so the range is [0,4]. The excluded input x = 0 would have output 1, but x = 2 still gives 1; that output is not lost.

Separate a definition from a plotted window

A graphing window can hide valid inputs or outputs without changing the function. State the domain and derive or justify the complete output set. A missing plotted point may come from a genuine exclusion or simply from the chosen display range.