Solve \(f \circ g(x) = g \circ f(x)\)
🎥 Video Explanation
📝 Question
Given: \[ f(x)=x^2,\quad g(x)=2^x \]
Find solution of:
\[ f(g(x)) = g(f(x)) \]
- A. \(\mathbb{R}\)
- B. \(\{0\}\)
- C. \(\{0,2\}\)
- D. none of these
✅ Solution
🔹 Step 1: Compute Both Sides
\[ f(g(x)) = f(2^x) = (2^x)^2 = 2^{2x} \]
\[ g(f(x)) = g(x^2) = 2^{x^2} \]
—🔹 Step 2: Equate
\[ 2^{2x} = 2^{x^2} \]
Since base is same:
\[ 2x = x^2 \]
—🔹 Step 3: Solve
\[ x^2 – 2x = 0 \]
\[ x(x-2)=0 \]
\[ x=0 \text{ or } x=2 \]
—🔹 Final Answer
\[ \boxed{\{0,2\}} \]
✔️ Option C