Evaluate \(f(g(h(x)))\)

🎥 Video Explanation

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📝 Question

Let:

\[ f(x)=\sin^{-1}x,\quad g(x)=[x^2],\quad h(x)=2x \]

where \([x]\) denotes greatest integer ≤ \(x\), and

\[ \frac{1}{2} \le x \le \frac{1}{\sqrt{2}} \]

Find \(f(g(h(x)))\).

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✅ Solution

🔹 Step 1: Find \(h(x)\)

\[ h(x)=2x \]

\[ 1 \le 2x \le \sqrt{2} \]

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🔹 Step 2: Find \(g(h(x))\)

\[ g(h(x))=[(2x)^2]=[4x^2] \]

Since:

\[ x^2 \in \left[\frac{1}{4},\frac{1}{2}\right] \Rightarrow 4x^2 \in [1,2] \]

\[ [4x^2]=1 \]

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🔹 Step 3: Find \(f(g(h(x)))\)

\[ f(1)=\sin^{-1}(1) \]

\[ =\frac{\pi}{2} \] —

🔹 Final Answer

\[ \boxed{\frac{\pi}{2}} \]