If `f(x)> x ;AA x in R.` Then the equation `f(f(x))-x=0,` has

Options

A. atleast one real root.
B. more than one real root.
C. no real root if f(x) is a polynomial and one real root if f(x) is not a polynomial.
D. no real root.

Answer

To solve the problem, we need to analyze the given condition and the equation step by step.

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that \( f(x) > x \) for all \( x \in \mathbb{R} \). This means that the function \( f(x) \) is always above the line \( y = x \) for every real number \( x \).

**Hint**: Consider what it means for a function to be greater than the identity function.

2. **Rearranging the Equation**:
We need to analyze the equation \( f(f(x)) - x = 0 \). Rearranging gives us:
\[ f(f(x)) = x \]
This means we are looking for values of \( x \) such that applying \( f \) twice returns \( x \).

**Hint**: Think about the implications of applying \( f \) twice and how it relates to the original condition.

3. **Applying the Condition**:
Since \( f(x) > x \), applying \( f \) to both sides gives:
\[ f(f(x)) > f(x) > x \]
This shows that \( f(f(x)) \) is also greater than \( x \).

**Hint**: Use the transitive property of inequalities to relate \( f(f(x)) \) and \( x \).

4. **Conclusion About Roots**:
From the inequality \( f(f(x)) > x \), we can conclude that \( f(f(x)) - x > 0 \) for all \( x \in \mathbb{R} \). This means that the left-hand side of our equation \( f(f(x)) - x = 0 \) is always positive and thus can never be zero.

**Hint**: Consider the implications of a function that is always positive in relation to finding roots.

5. **Final Answer**:
Since \( f(f(x)) - x \) is always greater than zero, the equation \( f(f(x)) - x = 0 \) has no real roots.

Therefore, the final answer is that the equation has **no real roots**.

### Summary of the Steps:
1. Analyze the condition \( f(x) > x \).
2. Rearrange the equation to \( f(f(x)) = x \).
3. Use the condition to show \( f(f(x)) > x \).
4. Conclude that \( f(f(x)) - x > 0 \) implies no roots.
5. State the final conclusion.