First slide

Functions (XII)

Question

The domain of the function $f (x) = \sqrt{\sin x + \cos x} + \sqrt{7 x - x^{2} - 6}$ is

Moderate

A

$[1, 6]$
B

$[1, \frac{3 π}{4}] \cup [\frac{7 π}{4}, 6]$
C

$[1, π] \cup [\frac{7 π}{4}, 6]$
D

None of these

Solution

$\begin{array}{ll} We have f (x) = \sqrt{\sin x + \cos x} + \sqrt{7 x - x^{2} - 6} \\ = \sqrt{\sqrt{2} \sin (\frac{π}{4} + x)} + \sqrt{7 x - x^{2} - 6} \\ f (x) is defined if \\ (i) 7 x - x^{2} - 6 \geq 0 \\ x^{2} - 7 x + 6 \leq 0 \\ \Rightarrow 1 \leq x \leq 6 & (1) \\ (ii) \sin (\frac{π}{4} + x) \geq 0 \\ x + \frac{π}{4} \in . . . . . . . . [- 2 π, - π] \cup [0, π] \cup [2 π, 3 π] . . . . . \\ \Rightarrow x \in . . . . . [- \frac{π}{4}, \frac{3 π}{4}] \cup [\frac{7 π}{4}, \frac{11 π}{4}] . . . . (2) \\ From (1) and (2), we get \\ x \in [1, \frac{3 π}{4}] \cup [\frac{7 π}{4}, 6] \end{array}$

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