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NSWMaths AdvancedQuick questions

Year 11: Functions

Quick questions on Domain and range for HSC Maths Advanced: the natural domain of a function (denominators not zero, radicands non-negative), reading the range off a sketch, restricted domains, and reading a graph backwards with horizontal lines, including parabolas and semicircles

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the natural domain?
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When a function is given by a rule with no domain stated, the convention is that the domain is every xx you can legally substitute. This is the natural domain. In Year 11 Advanced there are exactly two operations that fail for some inputs, and almost every domain question is one or both of them:
What is the range of a parabola from its vertex?
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A parabola is the most common range question, and you do not need to plot it point by point. Put the quadratic into completed-square (vertex) form y=(x−h)2+ky = (x - h)^2 + k, read off the vertex (h,k)(h, k), and use the direction it opens:
What is the range of a semicircle?
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A semicircle is the standard bounded example, and it is the reason square-root functions matter here. The full circle x2+y2=a2x^2 + y^2 = a^2 is not a function (a vertical line cuts it twice), but solving for yy splits it into two halves that are functions: y=a2−x2y = \sqrt{a^2 - x^2} is the upper semicircle (outputs ≥0\ge 0), and y=−a2−x2y = -\sqrt{a^2 - x^2} is the lower one (outputs ≤0\le 0).

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