f(x) = 31 2sin x 1 ≤sin x ≤ 1 2 ≥ 2sin x ≥ 2 2 ≤ 2 sin x ≤ 2 1 2 ≤ 1 2 sin x ≤ 1 +2 1 ≤ 1 2 sin x ≤ 3 3 1≥3 ...

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Definition of Functions

Range of fx...

Question

Range of $f (x) = \frac{3}{1 - 2 sin x}$

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Solution

$f (x) = \frac{3}{1 - 2 sin x}$

$- 1 \leq sin x \leq 1$

$⟹ 2 \geq - 2 sin x \geq - 2$

$⟹ - 2 \leq - 2 sin x \leq 2$

$⟹ 1 - 2 \leq 1 - 2 sin x \leq 1 + 2$

$⟹ - 1 \leq 1 - 2 sin x \leq 3$

$⟹ \frac{3}{- 1} \geq \frac{3}{1 - 2 sin x} \geq \frac{3}{3}$

$⟹ - 3 \geq \frac{3}{1 - 2 sin x} \geq 1$

$⟹ 1 \leq \frac{3}{1 - 2 sin x} \leq - 3$

$R a n g e = (- \infty, - 3] \cup [1, \infty)$

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