Evaluate \((f^{-1}\circ f)(1)+\cdots+(f^{-1}\circ f)(100)\)

📝 Question

Let \(f\) be an invertible real function. Evaluate:

\[ (f^{-1}\circ f)(1) +(f^{-1}\circ f)(2)+ \cdots +(f^{-1}\circ f)(100) \]

✅ Solution

🔹 Step 1: Use identity property

For an invertible function:

\[ (f^{-1}\circ f)(x)=x \] —

🔹 Step 2: Apply to each term

\[ (f^{-1}\circ f)(1)=1,\quad (f^{-1}\circ f)(2)=2,\ \ldots,\ (f^{-1}\circ f)(100)=100 \] —

🔹 Step 3: Sum the series

\[ 1+2+3+\cdots+100 \] :contentReference[oaicite:0]{index=0} —

🎯 Final Answer

\[ \boxed{5050} \]

🚀 Exam Shortcut

• \(f^{-1}(f(x))=x\)
• Reduce problem to sum of first 100 natural numbers
• Use formula: \(\frac{n(n+1)}{2}\)