Find \(f\circ(f\circ f)(x)\)
🎥 Video Explanation
📝 Question
Given:
\[ f(x)=\frac{1}{1-x} \]
Find:
\[ f(f(f(x))) \]
- A. \(x\) for all \(x\in\mathbb{R}\)
- B. \(x\) for all \(x\in\mathbb{R}\setminus\{1\}\)
- C. \(x\) for all \(x\in\mathbb{R}\setminus\{0,1\}\)
- D. none of these
✅ Solution
🔹 Step 1: Compute \(f(f(x))\)
\[ f(f(x)) = f\!\left(\frac{1}{1-x}\right) \]
\[ = \frac{1}{1-\frac{1}{1-x}} \]
\[ = \frac{1}{\frac{(1-x)-1}{1-x}} = \frac{1-x}{-x} = \frac{x-1}{x} \] —
🔹 Step 2: Compute \(f(f(f(x)))\)
\[ f\!\left(\frac{x-1}{x}\right) = \frac{1}{1-\frac{x-1}{x}} \]
\[ = \frac{1}{\frac{x-(x-1)}{x}} = \frac{1}{\frac{1}{x}} = x \] —
🔹 Step 3: Domain Restriction
Check where expression is defined:
- \(x \ne 1\) (original function)
- \(x \ne 0\) (division appears)
So domain:
\[ x \in \mathbb{R} \setminus \{0,1\} \] —
🔹 Final Answer
\[ \boxed{\text{Option C}} \]