Question

For the function \(f: [1, \infty) \rightarrow [1, \infty)\) defined by \(f(x) = (x - 1)^4 + 1\), among the two statements:
(I) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}\) contains exactly two elements, and
(II) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x + 1)\}\) is an empty set,
Options:

• A. only (I) is TRUE
• B. only (II) is TRUE
• C. both (I) and (II) are TRUE
• D. neither (I) nor (II) is TRUE

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a strictly increasing function \(f(x)\), the intersection of the function and its inverse, \(f(x) = f^{-1}(x)\), occurs on the line \(y = x\). For statement (II), the equation \(f(x) = f^{-1}(x + 1)\) is equivalent to \(f(f(x)) = x + 1\).

Step 2: Key Formula or Approach:
1. Check monotonicity: \(f'(x) = 4(x - 1)^3\). For \(x \geq 1\), \(f'(x) \geq 0\), so \(f\) is increasing.
2. Solve \(f(x) = x\) for (I).
3. Analyze \(g(x) = f(f(x)) - (x + 1)\) for (II).

Step 3: Detailed Explanation:
For Statement (I):
Solving \(f(x) = x \implies (x - 1)^4 + 1 = x \).
\((x - 1)^4 - (x - 1) = 0 \implies (x - 1)[(x - 1)^3 - 1] = 0\).
\(x - 1 = 0 \implies x = 1\).
\((x - 1)^3 = 1 \implies x - 1 = 1 \implies x = 2\).
The set \(S\) has exactly two elements \(\{1, 2\}\). Statement (I) is TRUE.
For Statement (II):
Let \(g(x) = f(f(x)) - (x + 1)\).
\(g(1) = f(f(1)) - 2 = f(1) - 2 = 1 - 2 = -1\).
\(g(2) = f(f(2)) - 3 = f(2) - 3 = 2 - 3 = -1\).
Consider \(x = 3\): \(f(3) = (2)^4 + 1 = 17\).
\(f(f(3)) = f(17) = (16)^4 + 1 = 65537\).
\(g(3) = 65537 - 4 = 65533>0\).
Since \(g(2)<0\) and \(g(3)>0\), there must be a root in \((2, 3)\). Thus, the set is not empty. Statement (II) is FALSE.

Step 4: Final Answer:
Statement (I) is true and (II) is false. Hence, only (I) is TRUE.