Problem

Use the parent \(\text{sine}\) graph to state its real domain, any excluded inputs, the reason for those exclusions or their absence, and its range as a separate set.

Exact sine graph over representative cycles with extrema visible; domain and range conclusions are withheld.

Big Picture

What this problem is really about

Keep domain and range tied to different questions: domain asks which horizontal inputs are allowed, while range asks which vertical outputs occur. Look for an expression that can become undefined when deciding input exclusions. For outputs, use the graph's attained top and bottom heights and include a boundary only when the curve actually reaches it.

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Its domain is all real numbers; excluded inputs are none; the reason is that there is no denominator; and its range is [-1, 1].

Its domain is [-1, 1]; excluded inputs are none; the reason is that sine is bounded; and its range is all real numbers.

Its domain is all real numbers except 90 + 180k, for integer k; excluded inputs are 90 + 180k; the reason is a zero denominator; and its range is [-1, 1].

Its domain is all real numbers; excluded inputs are none; the reason is that every trig function is defined everywhere; and its range is all real numbers.

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Use the parent \(\text{sine}\) graph to state its real domain, any excluded inputs, the reason for those exclusions or their absence, and its range as a separate set.

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