Sets, Relations & Functions
Range of Functions involving Square Roots
nta_pyq_2023_jan
Grade 11
Question:
The range of the function $f(x) = \sqrt{3-x} + \sqrt{2+x}$ is
\left[\sqrt{5}, \sqrt{10}\right]
\left[2\sqrt{2}, \sqrt{11}\right]
\left[\sqrt{5}, \sqrt{13}\right]
\left[\sqrt{2}, \sqrt{7}\right]
Step-by-Step Solution
Key Concept: Square $y=f(x)$ to get $y^2 = 5+2\sqrt{(3-x)(2+x)}$; maximize/minimize the product under the radical.
$y^2=5+2\sqrt{6+x-x^2}=5+2\sqrt{\frac{25}{4}-\left(x-\frac{1}{2}\right)^2}$. Max $y^2=5+5=10$, min $y^2=5$. Range: $[\sqrt{5},\sqrt{10}]$.
Correct Answer: 1