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SourceExam paper origin
CourseCurriculum level
DifficultyFoundation to Challenge
HSC.ADV.2001 Q5a Advanced
Development
Equations › Quadratic Inequalities Functions › Domain & Range
3 marks

State the domain and range of the function $y = 2\sqrt{25 - x^2}$.

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For the domain, the expression under the square root must be non-negative:

$25 - x^2 \geq 0$

Solve the quadratic inequality by sketching the parabola on the LHS and seeing where it is above the $x$ axis:

Quadratic graph

Domain: $\{-5 \leq x \leq 5\}$

For the range, $\sqrt{25 - x^2}$ has maximum $\sqrt{25}=5$ at $x=0$ and minimum $0$ at $x=\pm5$:

$$\begin{align*}0 &\leq \sqrt{25 - x^2} \leq 5 \\ 0 &\leq 2\sqrt{25 - x^2} \leq 10\end{align*}$$

Range: $\{0 \leq y \leq 10\}$

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