Find Functions \(f\) and \(g\)
🎥 Video Explanation
📝 Question
Given:
\[ g(f(x)) = |\sin x| \]
\[ f(g(x)) = (\sin \sqrt{x})^2 \]
- A. \(f(x)=\sin^2 x,\; g(x)=\sqrt{x}\)
- B. \(f(x)=\sin x,\; g(x)=|x|\)
- C. \(f(x)=x^2,\; g(x)=\sin \sqrt{x}\)
- D. cannot be determined
✅ Solution
🔹 Step 1: Analyze \(f(g(x))\)
\[ f(g(x)) = (\sin \sqrt{x})^2 \]
This suggests:
\[ g(x)=\sqrt{x}, \quad f(x)=\sin^2 x \]
—🔹 Step 2: Verify \(g(f(x))\)
\[ g(f(x)) = g(\sin^2 x) \]
\[ = \sqrt{\sin^2 x} \]
\[ = |\sin x| \]
✔️ Matches given condition
—🔹 Final Answer
\[ \boxed{\text{Option A}} \]