<p>Range of the function <i>sin</i><sup>–1</sup>(fog(<i>x</i>)) is</p>
<p>(A) <i>[</i>0, <i>π</i>/<sub>2</sub><i>]</i></p>
<p>(B) <i>[</i>0, <i>π</i><i>]</i></p>
<p>(C) <i>[</i>–<i>π</i>/<sub>2</sub>, 0<i>]</i>, <i>[</i>0, <i>π</i>/<sub>2</sub><i>]</i></p>
<p>(D) <i>[</i>–<i>π</i>/<sub>2</sub>, <i>π</i>/<sub>2</sub><i>]</i></p>
Step-by-Step Solution
Key Concept: To find the range of sin⁻¹(fog(x)), we must first determine the range of the composite function fog(x), then apply the inverse sine function which maps [-1,1] to [-π/2, π/2].
<p><strong>Step 1:</strong> Understand that sin⁻¹(y) is defined for y ∈ [-1, 1] and has range [-π/2, π/2].</p><p><strong>Step 2:</strong> For sin⁻¹(fog(x)) to be defined, we require fog(x) ∈ [-1, 1].</p><p><strong>Step 3:</strong> The problem asks for the range of sin⁻¹(fog(x)). Since the codomain of the inverse sine function is [-π/2, π/2], and fog(x) takes values in some subset of [-1, 1], the range of sin⁻¹(fog(x)) will be a subset of [-π/2, π/2].</p><p><strong>Step 4:</strong> When fog(x) varies over its entire valid domain where -1 ≤ fog(x) ≤ 1, the function sin⁻¹(fog(x)) traces through the complete range of the inverse sine function, which is [-π/2, π/2].</p><p><strong>Step 5:</strong> The inverse sine function is continuous and strictly increasing on its domain [-1, 1]:- When fog(x) = -1, sin⁻¹(-1) = -π/2- When fog(x) = 1, sin⁻¹(1) = π/2- For all values between -1 and 1, sin⁻¹ produces outputs between -π/2 and π/2</p><p><strong>∴ Answer:</strong> D</p>
Correct Answer: D