First slide

Functions (XII)

Question

$The range of f (x) = \sin^{2} x - \sin x + 1 i s$

Easy

A

$[\frac{3}{4}, 3]$
B

$(\frac{3}{4}, 3)$
C

$(\frac{3}{4}, 3]$
D

$[\frac{3}{4}, 3)$

Solution

$\begin{array}{l} f (x) = \sin^{2} x - \sin x + 1 = {(\sin x - \frac{1}{2})}^{2} + \frac{3}{4} \\ Now, - 1 \leq \sin x \leq 1 or - \frac{3}{2} \leq \sin x - \frac{1}{2} \leq \frac{1}{2} \\ or 0 \leq {(\sin x - \frac{1}{2})}^{2} \leq \frac{9}{4} or \frac{3}{4} \leq {(\sin x - \frac{1}{2})}^{2} + \frac{3}{4} \leq 3 \\ Hence, the range is [\frac{3}{4}, 3] . \end{array}$

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