If `f(x)and g (x) ` are two function such that `f(x)= [x] +[-x] and g (x) ={x} AA x in R and h (x) = f (g(x),` then which of the following is incorrect ? `[.]` denotes greatest integer function and `{.}` denotes fractional part function)

Options

A. `f(x) and h(x)` are identical functions
B. `f(x) =g(x)` has no solution
C. `f (x)+ h (x) gt 0` has no solution
D. `f (x) -h (x)` is a periodic function

Answer

To solve the problem, we need to analyze the functions \( f(x) \), \( g(x) \), and \( h(x) \) as defined in the question.

1. **Define the Functions**:
- The function \( f(x) = [x] + [-x] \), where \( [x] \) is the greatest integer function (floor function) and \( [-x] \) is the greatest integer function applied to \(-x\).
- The function \( g(x) = \{x\} \), where \( \{x\} \) is the fractional part function.
- The function \( h(x) = f(g(x)) \).

2. **Analyze \( f(x) \)**:
- For \( x \) being an integer, \( f(x) = [x] + [-x] = x - x = 0 \).
- For \( x \) not being an integer, say \( x = n + d \) where \( n \) is an integer and \( 0 < d < 1 \):
- \( [x] = n \)
- \( [-x] = -n - 1 \) (since \(-x\) is between \(-n-1\) and \(-n\))
- Thus, \( f(x) = n + (-n - 1) = -1 \).

Therefore, we can summarize:
- \( f(x) = 0 \) if \( x \) is an integer.
- \( f(x) = -1 \) if \( x \) is not an integer.

3. **Analyze \( g(x) \)**:
- The function \( g(x) = \{x\} \) gives the fractional part of \( x \).
- If \( x \) is an integer, \( g(x) = 0 \).
- If \( x \) is not an integer, \( g(x) \) will be the fractional part, which is between \( 0 \) and \( 1 \).

4. **Analyze \( h(x) = f(g(x)) \)**:
- If \( x \) is an integer, \( g(x) = 0 \) and thus \( h(x) = f(0) = 0 \).
- If \( x \) is not an integer, \( g(x) \) is between \( 0 \) and \( 1 \). Since \( g(x) \) is not an integer, \( f(g(x)) = -1 \).

Therefore, we can summarize:
- \( h(x) = 0 \) if \( x \) is an integer.
- \( h(x) = -1 \) if \( x \) is not an integer.

5. **Identifying Incorrect Statements**:
Now, we need to evaluate the statements provided in the question to find which one is incorrect:
- \( f(x) + h(x) > 0 \) has no solution.
- \( f(x) - h(x) > 0 \) is not possible.
- \( f(x) = g(x) \) has no solution.

Let's evaluate these statements:
- For \( x \) being an integer: \( f(x) = 0 \) and \( h(x) = 0 \) ⇒ \( f(x) + h(x) = 0 \) (not > 0).
- For \( x \) not being an integer: \( f(x) = -1 \) and \( h(x) = -1 \) ⇒ \( f(x) + h(x) = -2 \) (not > 0).
- Thus, the first statement is correct.

- For \( f(x) - h(x) \):
- For integers: \( 0 - 0 = 0 \) (not > 0).
- For non-integers: \( -1 - (-1) = 0 \) (not > 0).
- Thus, the second statement is also correct.

- For \( f(x) = g(x) \):
- For integers: \( f(x) = 0 \) and \( g(x) = 0 \) ⇒ they are equal.
- For non-integers: \( f(x) = -1 \) and \( g(x) \) is between \( 0 \) and \( 1 \) ⇒ they are not equal.
- Hence, there exists at least one solution (for integers).

Therefore, the incorrect statement is:
- **\( f(x) = g(x) \) has no solution.**